Is it possible to delete issues on GitLab?

It is now possible to delete issues starting with GitLab 8.6:

GitLab 8.6 released with Deploy to Kubernetes and Subscribe to Label

Delete Issues

Sometimes, simply closing an issue or merge request is not sufficient. For those times, we are now making it possible to delete issues and merge requests.

Only owners can delete issues by editing the issue or merge request and clicking, you guessed it, Delete.

At the time this question was asked, No. There is a feature request for that at: https://gitlab.com/gitlab-org/gitlab-ce/issues/2489

Now it seems possible: https://stackoverflow.com/a/36172116/895245

You cannot do that any more.

Delete existing issue (Deprecated)

The function is deprecated and returns a 405 Method Not Allowed error if called. An issue gets now closed and is done by calling PUT /projects/:id/issues/:issue_id with parameter closed set to 1.

DELETE /projects/:id/issues/:issue_id

Parameters:

id (required) - The project ID

issue_id (required) - The ID of the issue

Ref: GitLab Documentation