Appium integration with Teamcity
I have been looking for the same integration. There are two ways to achieve this.
Saucelabs is a Cloud platform to run the browser based or mobile based tests. There is a sauce labs runner that connects Teamcity with Saucelabs. Here is the link for that. https://saucelabs.com
Other option is to use NUNit runners to integrate Appium / Selenium with Teamcity using dot net frameworks. Here is the reference link that may help you get started. NUnit Plugin for Teamcity - https://github.com/JetBrains/integration-tests-vm-scripts.
While first option seems to be little costly even to start, second option did not have much documentation to start with.