Can I use 10\$\mu\$F caps in a MAX232?

They're definitely required! For their function look at the block diagram

You'll see that they're connected to the voltage doubler and voltage inverter. These create +10V from the 5V power supply and -10V resp. This is done by charge pumps.

An oscillator will control the switches so that either S1 and S3 are closed or S2 and S4. When S1 and S3 are closed C1 is connected to ground and V+ and charged to V+. When S2 and S4 are closed the top of C2, which is V+ higher than the bottom is connected to ground, so that the bottom now is V+ below ground. Via the switches the charge flows to C2, which will then have a negative voltage. So that's for the inverter. The same principle is used to double the incoming voltage.

C1 and C2 are the external 1\$\mu\$F capacitors. If they're omitted there won't be any +10V or -10V and no signal at the drivers' outputs.

In its datasheet Maxim recommends 1\$\mu\$F for the MAX232, 100nF for the MAX232A. I wouldn't recommend using 10\$\mu\$F instead of the 1\$\mu\$F. The switches might have some resistance and the capacitor may not get fully charged in the time one pair of switches is closed.

edit
In a comment to another answer which said 10\$\mu\$F will probably be OK someone said:

It should always be OK to use larger sized caps but they are required. The caps are part of charge pumps that produce and store +/- ~7 volts for RS-232 (I measured mine).

He didn't say, but it looks like he used 10\$\mu\$F. If you use the recommended value of 1\$\mu\$F you should get \$\pm\$10V. The 7V seems to confirm my doubts about the charging of the larger capacitors.

The purpose of the 4 capacitors is to generate the +/-12 volts needed for RS232 communications. The MAXIM and TI MAX232 datasheets recommend 1uF capacitors.

The MAX202 is a better choice, it can use 100n ceramic capacitors.