I bought a used bicycle with 28'' wheels. I would like to put 26'' wheels on it. Will that cause a whole bunch of problems?

That will mainly depend on what brakes you're intending to run. With rim brakes you'd have to bodge something up, no adaptors commercially available.
Disc/drum/coaster brakes won't be an issue - assuming the frame/fork can take them.

Apart from that it will put the pedals closer to the ground, so unless you're really good at coasting through curves you might suffer pedal strike.

If the bike have fenders it will look ridiculous.

It will lower the overall gear ratio a bit.

But why do you want to change the wheels?

Simply because I am short.

Overall you'd be best off to buy a mountain bike that already has 26 inch wheels and just put some road slicks on it if this is going to be a mostly road bike for you. Mountain bikes can be set up to fit vertically challenged folks a lot more easily than trying to swap wheels on a 28 inch bike. As mentioned the brakes would require a lot of fiddling and adapters for such a mod are not easily found.

1) If the bike is too high, then it's likely that it's too long too. Switching to a shorter stem might not take you all the way "home". Unless you already have the 26" wheels odds are that you'll just be pouring more money into something that never be anything but a poor compromise.
If you have the 26" wheels odds are that you have the rest of the bike too, which just might be a better option for you.
2) how have you determined that it's too high? Standover clearance? While it's nice to be able to have both feet on the ground while stradding the bike it's by no means required. The most critical feature is that you can remain in the saddle w/o tilting your hip (and a small bend to the knee) when the pedal is at its lowest. A bit of top tube interference can be overcome when stationary by leaving one foot on the pedal and leaning the bike a little.

You may be better off selling the bike you have and getting one that fits. You're treating the symptom and not the cause.

+100. It sounds like the OP has a fundamental fit problem. A Band-Aid approach will be both expensive and unsatisfactory.

how come you bought a bike that was to big in the first place? there are plenty of options out there with 700c wheels and small frames.

I have an awesome old 7 speed trek in like new condition with a tiny frame. how short are you? maybe we should trade ;-)

I have the opposite problem. Bought a used Boulder at the bike shop for my GF and I now know it is too small. 14" frame and she is 5'9. Beginner mistake and I think the LBS should have mentioned the size. Have the seatpost cranked up and a steerer extender so it's ridable to a degree. Friend says I need get rid of the knobby tires for her but I'm reluctant to spend any more on it because I know it will never fit her correctly.

Not saying that it isn't too small, but most MTBs these days have really sloping top tubes. That gives you a lot of exposed setpost even if the frame is the right size.

Yeah and it's a womans/girls version so it looks even worse. Her legs give out and it looks to me like she isn't getting the full extension. I'm afraid to raise the post any further. They let her ride it around the parking lot and she liked it. But that & doing 20 miles is a different story.

Not intending to further along what amounts to a thread hijack, but you can often find used parts for dirt cheap at an LBS (or at least you can at the ones near me). I got a super long seatpost for my slightly undersized MTB for $5, and a longer stem for $2. $7 and my bike fits. My frame is a 17.5", I would prefer something about an inch larger (and I'm about the same height as your girlfriend). Sounds like hers is a bit worse to say the least, but it's worth a shot. Just whatever you do, don't put the seat higher than the minimum insert mark. You're better off selling the bike on CL and possibly taking a loss than letting your ladyfriend get hurt by a broken seatpost.

To the OP:
Put your bike on Craigslist, offer it for trade for something smaller. Maybe you'll find someone who is in the opposite situation and everything will work out better all around.

I agree with all the suggestions to sell the bike and get one that fits. Even if it costs you $50-100 net, it will still be much, much less than:

$100-250 wheels
$50-80 tyres
$10 tubes
$100 fork
$25 cassette with taller gearing to preserve original gearing
$150-250 braze on cantilever posts for rear brakes
$150-250 painting
--------------------
$585-965 total

Just to install 26" wheels on a bike designed with 28" wheels. And the 1" difference in height may not be significant enough to make up for the improper sizing anyway.

There have been plenty of road bikes built with 650c wheels. Most are WSD, Womens Spcific Design, but if you are a male you could do a sex change on the bike with a man's saddle and maybe a color change if you don't like pink. Some of these bikes were built for the male population.
Bottom line is if you need smaller wheels you also need a smaller frame.

Not necessarily. We don't know the OP's height and inseam or what size bike they bought. A 5'6" rider may be a bit too short for the 57cm bike pictured below. But they can certainly trade it in for a 52-54cm model for a better fit and still use 700c wheels. You can get down to about 50cm and still have a flat top-tube with 700c wheels.

What size wheels/tires are on the originally-mentioned bike? 28" is not part of my bicycle tire vocabulary. Are these 700c/622? If so, where is the OP from that this is common terminology? There is also another 28" size, according to Sheldon.

28" (635mm) tires were common on older British bikes. They are stll in use on Dutch bikes.
700 c x38 tires are sometimes marked 1 5/8 x 28 just to add to the confusion.

So, we don't really know what size wheels/tires are on the OP's bike without more information, do we?