Dreamcast hardware (for a non cartridge based system) was pretty rock solid. MUCH better than original XBOX/PS2 hardware (disc read error issues/etc.). There are typically 4 things that can go wrong with a Dreamcast.
The first is the controller ports going out. Or, more specifically, the F1 resistor of the controller port going out which shuts down all 4 of the ports at once. The F1 resistor was put it to make sure there was not too much power running thru the ports (a type of stop gap fuse to protect the rest of the internal components). Based upon what I've seen/read/etc., you want to stay away from using Performance brand controllers as they were built with less strict tollerances and could cause the F1 resistor to blow out faster. Another controller to stay FAR away from are those X-Arcade real arcade type joystick conrollers. Almost every person I've heard using one has had the ports go out.
There are 2 fixes for the F1 resistor going out. One is to replace the resistor. The other is a bit of a hack, but has been found to work. Twist the F1 resistor until the leads touch, basically shorting it out and removing it from the equation. Since most people use regular (or well built) Dreamcast controllers, the issue of hurting something internal to the system (which the F1 resistor was used to protect) isn't much of an issue.
The second thing that can go wrong with a DC is the battery (as previously mentioned). The quickest way for a DC battery to go bad is to keep the system unplugged (like sitting in a closet). The battery obtains some of it's charge from being plugged in. Located on the controller port, and soldered into place, it takes a little tech knowledge/work to replace it, but can be done fairly easily. Or, you can simply reset the time/date each time you use the DC and keep on keepin on.
The third problem that typically occurs with a DC is not getting power. As long as the power supply has not burned out for some reason, there can be a very easy fix. Open up the Dreamcast, remove the power board, and clean the contacts. Over time, dust/etc. can cause them to get dirty/slightly corroded (oxidized rather than rusted) and lose contact. Simply cleaning them and replacing the power board usually fixes this problem.
The forth problem is the most serious. It involves the DC laser getting out of alignment. Most of the time this can be fixed but takes a good it of trial and error. Here are some links to how to adjust the DC laser...
http://www.whoneedsoxygen.net/index.php/2004/04/23/dreamcast-laser-calibration-trick/
http://www.consolevision.com/?Dreamcast:Tutorials:Fix_to_GD-ROM_calibration
The main thing to do when tweaking the laser is to LISTEN to the sounds of it trying to read the disc. The disc spins up, the laser gets into place (close to the center of the disc) then it raises and lowers (focusing the beam on the disc). If there is a problem the disc just seems to spin while you hear a sort of zip zip sound (laser assembly moving to try and grab hold of some data on the disc) but the disc will stop spinning and the game doesn't load. If the laser is able to obtain what it's looking for on the disc there is a VERY slight beep and the disc will start to spin faster (to load game data/etc.). THAT is what you are looking/listening for. You sort of have to work "towards" those results sometimes while tweaking the laser.
As for CD-Rs hurting the DC laser. I can sort of see where this could be seen as an issue as the games were originally on GD-ROMs. But, the DC was designed to be able to play music CDs and can even play VCDs. These are different formats than the GD-ROMs anyway. So, it's probably like anything tech related, simply using it as it was intended wears it down over time.