Tips turning brown are usually a sign of watering issues.
1. Make sure it gets a thorough soaking when you water it, so that excess water is running clear through the drainage holes.
Note: this does not mean immersion in water. Water from the top. If it takes multiple waterings to prevent your top soil from washing away, take your time until it runs clear.
2. Let it dry some in between waterings. Though azalea does like a more damp environment, few plants like it really wet all the time.
Note: Overwatering is just as bad as under watering. I generally see brown tips with overwatering, & limp or dry leaves when under watered.
3. The location of the bonsai may also be a factor. A tree in the sun will obviously dry out faster. One in a shady /partial sun spot will dry slower.
Note: Azaleas & Rhododendrons do very well in the shady/partial sun areas. Research growing habits of azaleas.
4. The top layer of soil dries fastest, but it doesn't mean the bottom is also dry. Gently dig down into the soil with your finger to check it's moisture. If you've got damp soil just below the upper layer, you're probably still in good shape. Give it another day, check again, & water.
5. If you're having issues with it not getting good water penetration (water not running free through the bottom), the holes may be clogged, or your tree may be potbound. Check the holes first. It's not a good time of the year to do any root work, but you can always take it out of the pot & place in a larger one with more soil around it.
Note: don't disturb the roots much if performing this task.
Also note: If this is your first tree, know that most have wires holding them in the pot (look under the pot). Cut them before trying to lift it out.
Also also note: keep it outside, even during winter.
Contact a bonsai club in your area for more tips. Good luck!