Does any Buddhist tradition have an explicit prohibition on participating in the rituals and practices of other religions? Are there any rules about entering other religions' holy places?
No, if we are talking about what the Buddha teaches in the Pali Suttas but one has to keep the precepts and practice in line with the Buddha's Teaching while participating in the other religion's practices.
In other words, the other religion's practices need to be compatible with the Dhamma.
In other words the other religious practice has to be dhamma.
I would say that in general, the various traditions usually frown on one practicing anything outside the tradition.
I would say that in general, teachers usually frown on it too because it can cause a lot of doubt, wrong view, misunderstandings and interference in the teacher's particular technique.
Yes, if these religious practices you want to participate go against the precepts.
ex: Animal sacrifice or slaughtering animals for food to celebrate a religious event.
Another is participating in worshiping Gods when you have taken spiritual refuge in the Buddha, Dhamma and the Sangha. It's like when you see a group of people who are boiling sand expecting it to be rice, you go participate in the same activity instead of showing them the proper way to cook rice.
Having said that, it is fine to participate in any activity that is in conformity with the Dhamma. ex: giving food to poor, releasing animals sold for meat etc. Also it is ok to take part in festivities as long as you don't get involved in activities that go against the Dhamma.