Are there any prohibitions on participating in other religion's practices? 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.