Do buddhists fall in love?
yes, certainly they do.
I'm new to the concept of Buddhism, and am failing to understand perhaps something very basic. I understand that one is not expected to hold anything close or dear as it is impermanent and could change at any moment so how does one love another person?
First, not all buddhists are fully and exclusively devoted to Buddhism, to it's fulfillment. This means that most buddhists live much like any other non-buddhist. They engage in all kinds of affairs of daily life, including work, marriage and so on. Others, like monks, likely renunciate romance in favor of their practice.
With that distinction in mind, we need some care when applying a teaching of the Buddha originally directed to someone fully devoted to the path, to a person who does not have the same commitment, disposition, preparation, etc. Otherwise, this could lead to something like taking a person who does jogging as someone who is (or should be) following a training program of an olympic coach (and worse, criticizing that person's performance on olympic standards).
"Romance" and passion are looked upon with caution, as they manifest quite explicitly strong craving and attachment. These are keywords of things buddhists often try in their mind to dodge. In the west, "romance" is a package of fantasies: find someone who is the source of one's happiness, the idea that the desire for each other is long lasting (or forever lasting), and often an implicit sense of ownership from both sides finds itself in as well....and all other things everyone knows are silly or unrealistic or simply unhealthy (say, from a psychological/therapeutic point of view).
On the other hand, an hypothetical/stereotypical non-monk buddhist may still have a companion and cherish that companionship. And their relationship might as well be called a loving one (the kind that is commonly said to be healthy: actively caring for their well being), even if lacking the aforementioned fantasies and delusions of romance, which are often born from our neurosis and fears.
The attitude this hypothetical buddhist would put an effort to have towards the companionship would be of focusing more on virtuous behaviors (of respect, kindness, faithfulness, honesty, etc) and relinquishing the blameful ones (jealously, dishonesty, disrespect, etc), bringing to peace strong unbalanced feelings. That is a good life, and according to Buddhism, one that may lead to a destination with less suffering.
In many traditions, "destination with less suffering" is what many buddhists look for when they do not set their current lives to shoot for nirvana. It's also how the Buddha "sold" buddha-dhamma in the pali suttas: if you don't feel like you can bear the dedication for nirvana right now, at least become a "good person": abandon most of the bad behaviors tied to greed, hatred and delusion in order to have less suffering in the future (and perhaps, find yourself in a position to finally go for it). That's a gist of his teaching: extinguish suffering, or at least lessen suffering.
"Therefore hold nothing dear, for separation from the dear is painful" - this seems to suggest that you should build a wall around yourself
The quote above is a strong statement about our desires and the nature of the objects we desire. It's an ultimate diagnostic of our condition, and an ultimate solution. But the actual method the Buddha proposed is not much building a wall between oneself and everybody/everything else (though a wall analogy can make some sense if it refers to seclusion from unwholesome states). It's about understanding these things in such a deep way that a natural dispassion towards them is developed -- like an adult who does not crave for child toys.
... and separate yourself from potentially wonderful feelings just because they could cause pain, and pain is bad.
It's not about separating ourselves from potentially "wonderful feelings". It's about having a deep understanding and appreciation for how these things are unable to provide everlasting shelter, and how feeble and really not that "wonderful" these "feelings" are -- when compared to other things, specially.
I think pain is a part of life; denying that pain, or ignoring it is not part of a healthy life. I think of this concept of Buddhism similar to "don't smell the flowers because you may prick yourself on a thorn" thus denying yourself of the delights found in the smell and also the growth / knowledge from the pain of the thorn prick. Both of which you can learn and grow from.
Buddhism is reasonably in agreement that "pain is part of life", and never "denies pain" or ignores it: suffering is a prime subject of study in Buddhism and, among many things, has been put in perspective how it supports the development in the buddhist path (SN 12.23) -- that is, suffering ends up being the reason for looking for an answer.
Pain and pleasure are part of our world. Fully embracing them as some sort of life ideology makes sense if one believes it will lead to either the same things (more pleasure and more pain, and this might be regarded as ok) or it will lead to a better existence -- e.g. one with less suffering.
In that case, if one believes one's life will lead to the same things in the future, and one is fine with it, Buddhism might not be much appealing.
However, if one hopes one's life/death will lead, eventually, to the permanent cessation of suffering, and a higher kind of happiness, then at least one can appreciate what the Buddha and the other ascetics of his time were trying to do. Many of them just did not believe this was the proper method for reaching this goal. Namely, that living a normal life delighting in pleasures of the senses and just trying to take a few lessons from the pain would lead to permanent cessation of suffering after this life (it could be said is somewhat in the direction, but it's not enough). In case of Nirvana as the ultimate destiny, the Buddha had an explanation for why this method is bogus and can't work. In few words: craving, in particular, for impermanent things as expedient for something permanent.
If one accepts all "these things around" as impermanent, than one is forced to accept that they cannot provide everlasting happiness. Thus, the idea that we can reach permanent happiness and escape from suffering by relying on "wonderful feelings" or delighting ourselves on mundane experiences of pleasure can be refuted.
This eagerness to have random experiences to fuel "wonderful feelings" into our senses, ultimately, ends up perceived as a distraction that seem to have no end, even though the feelings and experiences themselves do have an end, either exhausting themselves or becoming stressed by the sudden inability to get satisfied. Thus, any surprise on our part when facing changes around us are an indication of expectations disagreeing with reality -- in essence, delusion.
However, that does not mean buddhists should be cold and void as rocks. There are pleasures that are praised by the Buddha ("wonderful feelings" worthy of being cultivated), but all of them have a much lesser risk for craving (than sensual pleasures), and have a much better potential to help one out on a more sustained, healthier, wiser, pleasing life. So, it really depends what you refer to as "wonderful feelings", since these are scrutinized in detail in buddhism, where some are seeing as helpful, and some are not. Ultimately though (and the aspect you bring of buddhism is about the ultimate condition of things), they are still not the final answer since they are impermanent, even though they might be important to be cultivated -- to move towards a life with less suffering, or in order to fulfill the path.
Finally, it's worthy to note that there are many subtleties and important distinctions for a range of feelings and emotions in Buddhism, and these are very technical. For example, if one reads a buddhist text with the word "feeling" in it (translating vedanā), without proper background the reader's interpretation is likely to be off.