I have a Windows Phone 8.1 app which adds a tile to a Microsoft Band (version 1). Unfortunately it won't work because there are already too many 3rd party tiles added by other apps which I can't remove with the SDK (using the band client's TileManager.RemoveTileAsync() method)
There is a similar question discussing this topic here: Microsoft Band tiles which suggests that the Microsoft Health app has some way of getting a list of 3rd party tiles that may not necessarily use the SDK (I can't use the Microsoft Health app in this case).
Is there any way that I can remove 3rd party tiles using the SDK?
No, the Band SDK limits applications to interacting only with the Tiles that they, themselves, have created.
The Microsoft Health application has the ability to remove any 3rd-party Tile because, if it didn't, users would not have a means to do so if the application adding the Tile had, for example, been previously uninstalled.
If your application finds that the user's Band does not have space for an additional Tile, the application should instruct the user to use the Microsoft Health application to remove a sufficient number of Tiles and then return to your application to try again.