I would like to enter an open martial arts, MMA, or taekwondo tournament in my area. Any information on the whereabouts of such a location would be greatly appreciated.
For a second there from the way you worded your question, I thought you were basically without any knowledge of martial arts and just wanted to try entering a tournament just for fun to see how you would do. That would be hilarious! But I see from your bio that you're at an intermediate level in Taekwondo. That's better.
Okay, so as for open tournaments in or around Indiana, this should be an easy thing to search for on google. Google shows that the Indy Cup Taekwondo tournament is coming up in April. It seems to be a well respected and large mid-west region tournament. Probably open to all comers, but you'll have to check.
There are lists of tournaments that you can see on the web also. Like the ATA Taekwondo schools have a big web site for tournament listings, for example. Probably ITF, WTF, and the other TKD affiliations also have their own tournament listings. But the real question is whether they're for a specific school of Taekwondo or open to all. That's sometimes stated, and other times you just have to ask.
Probably your best bet is to follow a tournament circuit of some sort. Like NASKA. It can be either for Taekwondo specifically or for "sport karate". Circuits list their tournament schedule online. And these kinds of tournament circuits are often open to all competitors from all schools and styles. You just have to register for a sanctioned tournament near you, pay the entrance fee, and go and compete.
Also, since it's a circuit, if you actually win, you get points. The competitors with the most points at the end of a tournament season in a given minimum number of local, regional, and national tournaments have a chance to go on to the championship tournaments.
As for non-open tournaments, don't think those are off-limits to you. If I ever wanted to enter a tournament, and I knew it was closed to outsiders of whatever school system they have, I would just go to a local school (or email) and ask the instructor there if he/she can vouch for me and let me compete. It's sometimes just that easy.
They may ask who your instructor is, what your style is, etc. Or they may ask you to come to their classes to evaluate you in person. Don't think it's too awkward to ask. Anyone can ask. It's no big deal. The worst they can do is say no, laugh at you, or both. But there are a dozen other TKD schools in your area, and all you need is for one to say yes. You can meet a bunch of people and make new friends this way, also.
As for MMA... That's a different thing altogether. From your bio, I do not see anything resembling something that would allow you to have the ability to compete in an MMA match. These are matches, by the way, not usually tournaments, because in MMA people are generally too beat up to compete in more than one match per day.
There are cage matches and tough-man style competitions whereby anyone can enter, so long as they obey certain rules. The ones that let anyone compete are often pretty shady. Lots of drunk yokals yelling profanities at you, etc. You can do it, mind you. But like Dave Liepmann wrote, you need to know what you're getting into first.
Go to an MMA school. Do a sparring session with them. See how good you are there first. Chances are, you will be tapped out in short order. Unlike karate and TKD schools, you can get very hurt competing in a full contact MMA match.
If you go to this web site, it lists MMA events:
You might also want to read this:
"How to find MMA fights" http://www.mmarecruiter.com/index.php?do=/forum/becoming-a-fighter-28/how-to-find-mma-fights-2/
MMA events, unlike the rest you mentioned, are generally not tournament-based, but rather based on planned-well-in-advance ring (cage) fights between pairs of specific fighters. Maybe an MMA promoter near you can schedule you a fight.
Before entering a full scheduled ring fight, it would be a good idea to go to an MMA school and ask to have an MMA-rules sparring match. It might be eye-opening.