Honestly, it doesn't sound like it's infected. Especially if it's not hot or itchy. Just sounds like it's still in the healing phase, it's normal to have some discharge and whatnot as the body heals-just like it's normal for a cut or something to ooze a bit and scab over.
If you don't want to take it out, don't. Because if you do, and you are determined to have your belly button pierced, you'll have to start the entire process all over again.
Two things though:
1. Go to the doctor and let them take a look-just to be safe. You can leave the piercing in for this.
2. Change your piercing to a high quality pure metal (like gold). Some people have reactions to low-quality or nickel plated jewelry. Sounds like this could be what's happening to you. When you change a fairly new piercing, wash the area and your hands. Use pure vaseline to get the new jewelry through without catching or damaging the piercing/tender skin. Rinse all the vaseline away after. This is what I do.
If you don't want to take it out, don't. Because if you do, and you are determined to have your belly button pierced, you'll have to start the entire process all over again.
Two things though:
1. Go to the doctor and let them take a look-just to be safe. You can leave the piercing in for this.
2. Change your piercing to a high quality pure metal (like gold). Some people have reactions to low-quality or nickel plated jewelry. Sounds like this could be what's happening to you. When you change a fairly new piercing, wash the area and your hands. Use pure vaseline to get the new jewelry through without catching or damaging the piercing/tender skin. Rinse all the vaseline away after. This is what I do.