Suppose I reconfigure my HTTP server to add the following HTTP header into every reply:
or some other presumably funny text.
Can this have any negative effect on the server security?
No. It has absolutely zero effect.
It depends what the header is.
Assuming it is a made-up,
X--prefixed, custom header, with a static value, then @Polynomial's answer is correct - it has absolutely no effect.
If you build the header value based on something else - e.g. the user's password - then that obviously would be A Bad Thing™.
As long as you don't disclose info potentially valuable to a would be attacker you are fine.
X-our-awesome-stack: Software vX.YZ, Anothersoft vX.YZ, we dont believe in input santiziation! could disclose info that you might not want to advertise.
X-Hello: What's up type of headers are absolutely harmless.