For example, when there is only one remote: https://gitee.com/a/b.git, you can't open anything on remote.
If you have two remotes: https://github.com/a/b.git and https://gitee.com/a/b.git, you can open the gitee remote, but the url is github. The key point is, there must be an origin of github.
See the gif:

Does this still happen with GitLens 11?
Yes, it does.
After clicking "Open Repository on Remote":

I believe the issue is that GitLens can't auto-detect your gitee.com remote, so it needs to be configured first. I have pushed a "fix" for this in the insiders edition of GitLens -- where you'll get a better error message.
Can you please try the insiders edition? Be sure to disable/uninstall the stable version of GitLens first.
You can install the insiders edition from here.
Sorry, I don't have much time for testing. You say GitLens can't auto-detect the gitee.com remote, but obviously it "knows" what the remote url is (just use fetch or push). I don't know what "detect" here exactly means, but user just want to open the remote url in browser quickly, rather than an error message.

GitLens knows the git remote "url", but that doesn't provide enough information to be able to open a specific resource (branch, commit, file, etc) given that url (never mind that the git url could be git: or ssh: rather than http). So in order for GitLens to be able to open those resources it needs to know how to form those urls -- and those need to be added via the gitlens.remotes setting.
I get it, thanks for your reply. I don't have time to test it, but as long as the error message points to gitlens.remotes, it should be OK.