For context of my answer to this thread's questions. I am half-Jewish in the sense that my mother was Jewish and me Father was a christian. I decided to partake in the Jewish traditions and identify it as my personal choice in religion. My mother's side of the family is either dead or has converted to Christianity, my father's side varies from average christian to fundamentalist Christians and the area I grew up in was heavily, heavily catholic. To add to it I actually had one Buddhist friend and one Muslim friend in my life as well, along with many Atheist or Agnostic friends.
There is a time, a place, and a how and it depends on your religion AND the other religion in question. Some religions may say it is okay to follow some traditions of other religions, others may say that it is not. If you are not sure ask someone who practices that faith, study it or if the problem is your uncertainty on whether or not YOUR religion accepts it, than ask your religious leader or study on your own religion to find out the answer. Education is key to this though, and is very helpful when coming to respecting the other's religion.
Like, say, a Catholic travelling in Japan washing his hands in a shrine before entering a Buddhist temple. Or a Buddhist partaking in Communion.
Should that be allowed? or should you say "I'm sorry, but, I'm Catholic." or whatever, and not have to partake in Communion or wash your hands in front of the gods.
There is a major difference between signifying devotion and following traditions out of respect. The best example is Hinduism, which you are not allowed to walk into their temples with shoes on, but instead barefoot or with a form of sandal called "Khadau" because by their religious beliefs you can NOT make the temple floor dirty, out of respect to their god's place.
At the same time, taking the communion and NOT being of christian faith is actually insulting in it's own right because it is a sacrament of devotion to Jesus, eating his 'skin, blood and soul'. If you don't believe in the christian faith and you take it, you are treating it like it's just some average wine and bread and not like it's an important holy symbol... especially if it is well known that you are not of their faith. Respectfully declining and stating that you follow a different religious belief and thus can't is actually more commonly looked up upon than down since you're respecting the fact that it IS a sacred religious tradition for only the believers of that faith.
Before you EVER go inside another temple of a religion that is different than yours, either have a friend of that temple, who can explain the traditions and why they are what they are to you so you know what you should or shouldn't decline, or study up on the religion. There are times when it is VERY disrespectful to follow sacred or devotion traditions of a temple/church/shrine and there are times when it's even more so to NOT do such. Knowing the difference between the two is of great importance.
I was listening to an NPR piece this morning where they were in Japan and the reporters had to wash their hands in front of the gods before they went up to some temple, and I was just wondering, like, what if you were Christian or Jewish or some other religion. Would you still have to? Can someone of another religion wash their hands, or would it be a sin?
Religion by nature is it's own tricky subject, despite it being a set of beliefs. It depends on YOUR religious beliefs and the set of values you have. If you do not know you should look into the other religion and then into your own and determine this yourself. Most of what I have to go by is from personal experience and from my own limited knowledge of other religions. For this question though, I would say yes you can do it to show your utmost respect to the religion since it is doubtful that you'll be breaking a commandment/Mitsvah/Rule of your religion.
Many religions DO have a 'safety net' to fall back on for when you DO sin, knowing your religion's version of these is important since if you go into a religion's god(s)' place, there is a chance you might have to sin just to be respectful towards this religion. At the same time, if you go into a religion's god(s)' place, you need to know that your religion is not theirs, and thus try everything in your own power to stay the utmost respectful in many of the holy place's traditions. ALWAYS pay attention to the main details of the traditions, their meanings and their importance. If a tradition is considered sacred and should only ever be done by the practitioners of that faith, than do NOT participate in it for you'll in turn insult them. at the same time other traditions of importance, like washing your hands, or taking off your shoes or things like that are to be done (Otherwise don't even bother.)