One possible solution that does not require configuring a server, is to upload the images to a free service like imgur (unlimited API uploads for open-source projects) using credential stored in a "Travis secret", see https://docs.travis-ci.com/user/encryption-keys/.
The main problem with this solution is that these secrets are not accessible during Travis builds that are caused by pull requests generated from forks. Only pull requests from the same repository (but from a non-master branch, naturally) can use these secrets. (See https://docs.travis-ci.com/user/pull-requests#Pull-Requests-and-Security-Restrictions.)
If testing only post-merge commits or pull requests from the main fork is OK, then that Travis secrets + imgur is a viable option.
@SchrodingersGat: Unfortunately the GitHub API does not allow you to upload images. GitHub internally receives the images, uploads them on the fly and then the brower-side interface inserts a link into the message.