Is it normal for my dsPIC to run warm?
You have all the information, so do the math and you can answer the question yourself.

The PIC is at 37C and ambient is at 23C, for a rise of 14C.  The first knee jerk reaction is that this is quite reasonable.

Next you look in the datasheet and see what the thermal resistance to ambient is.  This is shown quite clearly in table 25-3 on page 272, right where you'd expect to find it.  It says the typical thermal resistance of the 64 pin TQFP package is 40 degC/Watt.  Since you see a 14C rise, that implies the chip is dissipating 350 mW.

Next you look again in the datasheet (do you see the recurring theme here?) and see how much power you should expect the part to dissipate.  Table 25.5 on page 274 show the power current at 3.3V and several operating frequencies.  The maximum is 90 mA at 40 MIPS, which is 297 mW.  That's a bit less than it is apparently dissipating, but the 297 mW value doesn't include any of the peripherals or extra dissipation due to I/O pin current.  Overall, the numbers look close enough and there is enough slop so that your observed temperature rise doesn't sound out of line.