I have done this twice, once on a Toro Recycler and on a cheapo MTD mower, in both cases the results were a significant improvement.
I did it by using some soft wire, think small twist tie with the plastic removed, to tie together a few coils on the governor spring. This allowed the governor to still perform its function, but the slight bit of added resistance on the spring brought up the RPM nicely.
I am not sure how much of a change I made on the Recycler model, as I never checked before and after, but gave it away to a family member in need of a more powerful mower a few years and about 100hrs of use later. They still have it today and it still runs well, but sees much less use and zero maintenance. I always ran it on M1 5w30 with annual changes.
The MTD has a small 125cc engine that was quite anemic in anything but thin grass and did not cut well even with a sharp blade. I used a tiny tach on this one and raised the RPM from about 2700-2800rpm up to about 3400 rpm. It now cuts well and has plenty of power to go through the thicker areas without having to rock it back and forth or pass over twice.
Other than it sounding like it was going to take off compared to running it at the original RPM, I have noticed that it seems to run about 20 - 30 degrees hotter oil temp using an IR thermometer on the back of the engine block after about 30 minutes of mowing, from about 230F to about 250F - 260F now, so I switched it to a straight 30w oil from the 5w30 I was running. I have not tried using it for a longer mow, as I have a riding mower now to do the larger areas, but it has been running like this for about 3 years now, probably about 25hrs of use. Always starts on the first pull.
As I believe was mentioned, its a good idea to calculate your blade tip speed. I went with 3400 on the MTD based on a 19" blade to keep the speed to about the recommended 17k fpm for commercial mowers. FWIW, I am not condoning raising the RPM, nor would I want to exceed a 17k or 18k fpm blade tip speed for safety reasons, just sharing my experiences.