I flew B-1B's for 7 years. I've also had flights in F-15s and F-16s. The B-1 has 4 afterburners, but a lot more gas than the fighters, so I rarely had to stay out of burner due to fuel. There's lots of reasons to minimize burner usage, though:
1) Operationally, AB makes you highly visible to all. At night, you put a spotlight on yourself. Daytime, everyone on the ground can hear you. IR sensors will find you quickly and easily, and even lower tech-IR missiles will prefer your burner to flares.
2) That extra 50% beyond mil power is actually an awful lot. When you do use burner, you don't need it for long. The B-1 could accelerate in full AB from .8 to .95 mach in just a few seconds. Operationally, you just don't need AB that much or often. If you're trying to defeat a missile, you're going to use excess airspeed first to slow to cornering velocity. The B-1 can maintain cornering velocity without burner since it's at relatively low g. A fighter at 7+ g will need some burner to maintain energy especially at cornering velocity, but as it can turn 90+ degrees in just a few seconds, it doesn't need much or any burner. Regardless, in a turn to defeat a radar missile, since IR missiles detect 'passively' meaning there is little or no warning, a pilot will frequently assume there's a heat seeker in the air when turning to defeat a radar missile and will avoid burner anyway.
3) Close-in air to air dogfighting is one of the few times a combat aircraft needs extended burner. In fighter combat, energy management is very important. No one wants to be on the losing end. Get too low on airspeed, and your jet turns too slow and you lose, so fighter pilots will use whatever burner they need to keep the threat off theier tail and win the fight. In the B-1 as well, in fighter intercept exercises, that was when we tended to use more burner. We tended to use it to accelerate quickly to complicate the fighter's intercept, and in some cases to bug out with a fighter on our tail.
4) The other regime where burner use is frequent is takeoff. This is statistically one of the most dangerous phases of flight, and reaching flying airspeed quickly minimizes the danger. When I was flying, the B-1 always took off in burner -- not sure now. Fighters can under certain conditions take-off in mil power, but I've rarely seen it.
5) Burner use in American jets ABSOLUTELY DOES NOT ADD SIGNIFICANTLY TO REQUIRED MAINTENANCE AND DOES NOT HARM THE ENGINES. The poster who mentioned that might have seen something on the MIG-25, which will destroy its engines in high speed flight. Presumably, other soviet fighters have some maintenance issues with burner use, but American combat aircraft are built to employ burner whenever needed without damaging engines.
6) Altitude is a very important point, as burner fuel flow will decrease with altitude. In thin air, there is less oxygen available for combustion, so the fuel controls have to adjust accordingly. As the previous poster wrote, thinner air creates less drag making it easier to go fast. But... as a commercial pilot today, I've flown with many former fighter pilots, and whenever we get to talking about it, few of us have spent time above 40,000 feet. The higher service ceiling is a nice stat for the contractor sales teams, but there's rarely an operational reason, and a lot of bad things can happen (like engine stall and physiological emergencies) up in the 40's.