Confirming what you've surely always known deep down, the vast majority of those buttons you push at intersections do not do anything to shorten the amount of time you are going to wait to cross the street. SFGate, inspired by a piece in the Boston Globe that revealed that Boston employs total placebo buttons for pedestrians in a majority of downtown intersections, reached out to the SFMTA to see whether the same was true here. And the answer, essentially, was yes.
In Boston, officials decided some years ago that "the city’s core is just too congested with cars and pedestrians to allow any one person to manipulate the cycle" with a button, according to the Globe. The buttons are still functional in some outer parts of the city where there's less traffic, but otherwise all the traffic signals are controlled by timers that fluctuate throughout the day and the buttons still sit there on traffic poles as relics of the past, having been installed at a time when traffic was lighter and pedestrians could ask for the right of way.
In San Francisco, of the 1,222 intersection crosswalk buttons across the city, only 251 are "pedestrian actuated" according to SFMTA spokesperson Paul Rose, but as he spins it for SFGate, "In San Francisco, every push button has a purpose." That is to say, none of them are there just to be placebos, but most are actually just there to help the visually impaired. Those buttons that don't actually effect when the light changes do produce what's called an Audible Pedestrian Signal (APS)," often that little voice that says "Walk!" and then makes that grating jackhammer sound to let the visually impaired know they can enter the intersection.
There are also plenty of intersections in SF where they've just done away with the buttons altogether, because we all know the drill.
SFGate provides the following map below showing where the functioning, pedestrian-actuated cross signals are, and as you can see a lot are clustered along the long, busy thoroughfares of Third Street, the Embarcadero, and 19th Avenue where car traffic tends to outnumber pedestrians who need to cross.
This guy on Flickr wrote a lengthy treatise on why actuated signals are actually bad for pedestrians, noting that having timed signals that switch over on regular one-minute intervals are likely to get you across a street faster than, say, the Embarcadero where you might stand around waiting for the system to acknowledge you for god knows how long.
And, naturally, here in SF, pedestrian advocates Walk SF have something to say about these buttons, which they refer to disparagingly as "beg buttons." They support getting rid of them unless they are necessary for the disabled, saying, "We don't think people should have to ask to cross the street."