Is it weird to display ads on an academic website? I have never seen (or realized) an example of an academic website which displays ads.
I am thinking of a website not hosted by the institute, but as a personal website, paid by the faculty member.
Does it look weird/unethical/unacceptable if a person uses a service similar to AdSense for their website?
Such a website is itself an ad: for the lab, the lab's research, and for the personnel running/working at the lab.
In many ways, a lab website is like a resume. While it's not unusual for newsletters published by some local org (say, a school, church, youth group, etc) to have local ads to cover publishing costs from the neighborhood dental clinic, grocer, etc, but you wouldn't expect to find these on a resume.
Probably most people are conditioned enough to online ads that they wouldn't even notice them, but if they did it might seem a bit weird, and if they were at all intrusive it would be downright ugly looking and reflect badly on the lab. If I were working in a lab where the PI had a website with ads, I'd be embarrassed by it even though it wasn't my own responsibility. If I was visiting such a site, I might wonder if I've stumbled on some illegitimate content farm site by mistake.
I doubt a typical academic's website gets anywhere near enough traffic to make ads make much sense financially. The cost of potentially putting off a precious visitor doesn't seem worth it to me to pocket a few pennies.
If the site is hosted by the university or with any grant funds for research it is likely illegal or against some terms to make any personal profit off such a site; even if the funds are just used to pay back the site costs it's a gray area that is still probably not allowed. A university could certainly even have a policy on this even if their funds are not directly involved if anything about the website affiliates it with the university, including being an official or pseudo-official website for a lab operating out of that institution.
It's just not worth the potential conflicts from any angle I can see.