To make TeamCity build pull requests for a repository hosted by Gitea you have to add the following line to the “Branch Specification” field in the VCS settings of the repository on the TeamCity side:
Afterwards TeamCity will build each pull request using the logical branch name “pull/X” (where X is the number of the pull request).
Sadly there is no tighter integration (yet) to publish commit build status. But maybe we can figure that out too later on. ;-)