TLC5940 - with 200LEDs from car battery
Regarding resistors, the TLC5940 uses a constant current sink driver for all 16 LED drive lines.  That baseline current is set by one resistor, sec 6.7 figure 1.  There is a dimming function provided by the PWM capability.  Note that for color balance RGB leds are tricky in that if all three LEDs are getting the same current the light may not look 'white' to the eye, so that could get complicated, especially since that reference resistor is applied over all 16 lines, so you either use PWM to rebalance, or you run each R/G/B color to a separate TLC5940 and tweak the reference resistor.

One thing to watch on the datasheets notice how that DC forward voltage changes for each LED (even within a batch of LEDs with min/typ/max). Color and luminosity grades (binning is the vernacular) can become critical in high end applications (don't know what your's falls under). the 3.4V is the typical white LED, note the 2.1/3.1 typicals for the R/GB, plus the spread.  A 53 ohm resistor would dump
60mA into the red LED (typical).  Thats' why the current sink technique on the TLC5940, brightness is proportional to current.

From the LED datasheets you should set your current to 20mA or lower for best LED reliability, so this indicates a reference resistor of about 2K ohms.

Regarding TLC5940 overheating, you need to pay attention to section 11.2, 11.3 as well as sections 9.x.  

Figure 3 gives you an idea of the power dissipated in the chip by the current sinks, so ,02A * 0,5V *16outputs = 0,16W per chip, which isn't bad, for a total of 4W dissipation.

You'll note that the different packages have different maximum dissipation rates section 6.7 fig 2, -especially- if the power-pad heat slug is not soldered down properly to a very good copper plane to conduct heat away from the chip. Notice the 2:1 difference, and note that you will not be able to solder that powerPAD with a regular soldering iron- it is underneath the chip body.  You will require at a minimum a hot air soldering station or better yet a friendly business with a reflow oven.

Luckily at 20mA you're ok, -but- say you decide you want to PWM the LEDs, then you may want to change the maximum current drive to higher level and that power dissipation factor may become critical, especially depending on the ambient temperature.

I recommend reading that datasheet and and make sure you understand what and why they are doing.  Notice that the LED drive inputs can take up to 17VDC, so that means that you can also do strings of LEDs, which may make your life easier (until one burns out and the string goes dark).