No matter who uploads video content to YouTube - be it an individual, another site or a third-party app - the uploader is still bound by YouTube's Terms of Service and copyright stipulations:
You further agree that Content you submit to the Service will not
contain third party copyrighted material, or material that is subject
to other third party proprietary rights, unless you have permission
from the rightful owner of the material or you are otherwise legally
entitled to post the material and to grant YouTube all of the license
rights granted herein.
So if one of your users uploads video content (to your site and which is uploaded to YouTube) that is copyrighted by someone else, they are liable, and, generally speaking (depending on jurisdiction) not you.
But it's more possible YouTube would be unhappy that you are automating uploads:
You agree not to use or launch any automated system, including without
limitation, "robots," "spiders," or "offline readers," that accesses
the Service in a manner that sends more request messages to the
YouTube servers in a given period of time than a human can reasonably
produce in the same period by using a conventional on-line web
That could be an issue, unless you arrive at an agreement with them to allow you to automate uploads.