A temperature spike like this can be one of two causes:
a) CPU is overheating
b) temperature gauge is reporting the wrong figure
The first will occur if the CPU is being worked hard, and your heat sink/fan is incapable of handling the job, or seated incorrectly. The latter is more likely (most heat sinks sold with computers are up to the job, if installed correctly).
Which is the case is easy enough to test with your fingers - run the computer with the case closed but not screwed down, and when you hear beeping, remove the side of the case and put your fingers NEAR BUT NOT TOUCHING the heat sink. It may burn you if it's too hot. It _should_ be cool enough to touch (at least briefly) - but then again, yours is potentially overheating.
To cause the overheating problem, run a CPU burn-in program, or a program which utilizes 100% of your CPU (my favorite is SETI@home - which also assists the Search for Extra-Terrestial Intelligence - see
http://setiathome.ssl.berkeley.edu/download.html). This should tax your CPU enough to cause a warning beep.
Your case temperature should rise along with your CPU temperature unless your case is VERY well ventilated - but not nearly as much. A 20-30 increase in CPU temperature may cause 1/10th or so as much increase in case temperature. The case would also likely be slower to cool down as case cooling is generally more passive and/or less efficient that CPU cooling.
Dust in the heat sink would reduce performance a bit - if the dust build up is really bad it coud be substantial. This should be easy enough to tell just by looking at the heat sink and fan. Are they COATED with dust, or just a little dusty? A little dusty is not really a problem at all.
If you do discover that your CPU could use better cooling, and you want to try the cheap option first, buy some arctic silver paste and visit
http://www.arcticsilver.com/instructions.htm for instructions on how to use it. It will involve removing the heat sink from your CPU, cleaning it, and pasting it up with the thermal paste before remounting the heat sink. This will increase the temperature flow between your CPU and the heat sink, reducing the CPU's temperature. If this doesn't work, you'll need a new heat sink.
If you discover your temperature gauge is at fault - you have a few choices. Reflashing the BIOS with another version may fix it (BIOS upgrades sometimes re-calibrate reporting of the temperature, when found to be innaccurate). If not, you may have to disable the heat warning for the CPU. This is in itself not the best path to take - unless there is a separate warning for the CPU fan failing, if the fan stops your computer may melt (literally). This is not a very common occurance however.
Don't be too exasperated Temporal - its a matter of finding the problem (by confirmation or elimination) and then fixing it. Most PC problems you can fix yourself given a little know how and a little work.