I work in 911 dispatch, in my jurisdiction I wouldn't exactly call this a major problem most of the time, just a minor annoyance. That said, we do get a lot of butt dials.

Policies vary on this a lot from one jurisdiction to another, where I work for open lines and hang-up calls from a cell phone, we listen to it for about 30 seconds and if we don't hear anything suspicious we hang up and disregard it. If we end up getting multiple calls from the same phone number we'll usually try to call back and if we have a decent fix on the location, send an officer out to investigate.

I know at least one of our neighboring counties does send an officer to check out every cell phone hang-up, I'm pretty sure that they pretty much just drive through the area and as long as there's no obvious emergency they clear it out, but even still if they get half as many as we do I don't know how they manage to keep up with it in addition to actual emergencies.

If you find yourself accidentally dialing 911, best thing you can do is just stay on the line, and say "sorry, this was an accidental dial, no emergency" and as long as we don't hear anything suspicious most places will disregard it. You don't need to explain everything that's wrong with your phone or what you were doing, etc. no offense, we don't really care, at that point we're either going to dispatch police or not depending on what we hear and our agency's policy, and those kinds of details aren't going into our notes. It's not a big deal, we get probably hundreds of them a day. That applies extra if you accidentally call from a landline, pretty much everywhere will send an officer out to investigate a hang up from a landline since we have a fixed location to check)

Some areas have very high 911 call volumes, damn-near every dispatch center in the country is always short-staffed (some worse than others, mine isn't too bad, but in the almost 5 years I've been here I don't think we've ever been fully-staffed, so if you're interested your local 911 center is almost definitely hiring,) and of course some places have a lot of actual emergencies going on, so if you can avoid accidentally calling, definitely try to, you may only be taking up a few seconds or minutes of a calltaker's time, but if they have staffing issues or a high volume of calls, that could still mean delays in someone else getting a response to their actual emergency.

Some people like to call back after they hang up to let us know it was an accidental dial. Most of the time I'd say it's not totally necessary and just taking up our time twice. If it's from a cell phone, like I said, a lot of places are going to totally disregard it, and if it's from a landline usually the officers are going to stop by to verify everything is OK anyway (we will let them know you called back, but at that point it's out of our hands and up to their discretion) so kind of use your best judgement about if you should call back or not.

Also, for those not aware, a deactivated phone (with no service, sim card removed, etc) can still call 911. A lot of people give their old phones to their kids to play with, personally I think that's a bigger issue. If they accidentally call us once to them that just means they found a new thing their toy can do and they're probably going to call us multiple times, and because they're playing or may not understand what's going on they may tell us they have an emergency when they don't. We also can't call those deactivated cell phones back to try to speak with an adult to confirm if everything is OK, so depending on what the kid tells us you may get police, fire, and/or an ambulance showing up at your home.