Usually no, but if the institution bans certain religious displays, such as Christmas trees, manger scenes, or displays of other cultures then it is not being tolerant. It is instead being intolerant to those that were not allowed to display their symbol or display. Intolerance of everyone does not equal tolerance. Too often an institution attempts to please everyone by not allowing anything. That is not tolerance or inclusivity. It is exactly the opposite.
We live in a melting pot where there are people from all walks of life with a multitude of different beliefs. In order to show true gratitude one has to show tolerance and appreciation for others, along with their traditions. It truly is a part of growing up and being an adult. You enrich yourself when you learn from other cultures so I don't see how one would equate that with unintentional intolerance. It is good to step outside of your comfort zone and embrace the unknown and what is different in ourselves and others. Celebrating the holidays is definitely a way to do so.
Being tolerant of all holiday practices is merely being tolerant of other people in the world. This couldn't reasonably turn into intolerance based on the fact that one showing tolerance to practices not one's own simply means respecting those around you. Tolerance does not reflect understanding or desire to practice something one does not believe in, but rather making space for those practices that are different then your own.
All holiday practices and beliefs stand alone on their own merits. Christmas and Chanukah are not mutually exclusive holidays. Whatever any person believes is entirely of their own choosing, and to be tolerant means to allow that to happen. The beliefs of one group do not need to be shoved in the face of everybody, just as not believing or celebrating a particular event does not mean a person should take offense that others do.
No, I do not believe that tolerance of all holiday traditions inadvertently leads to intolerance of people or their beliefs. It is my opinion that learning about and even participating in various other holiday practices promotes a healthy community and educated learning. There's so much that others do for the holidays that people aren't being told about. People would rather enhance the negatives than support the positives. Even complete tolerance gives light to the fantastic differences of people. I believe that the more people become involved, the more they'll accept that intolerance is a bad choice to make.