How exactly does one calculate the proper filter capacitors for the 567 tone detector?
Making it go:

It is likely that the lack of output is due to you not having allowed for the fact that the LM567 has an "open collector" output.
   (Rl on pin 8 in the diagram below. Output is low when tone is 'detected'.)

It needs a resistor or similar to V+ to operate.
Without this the device is about as good at detecting tones as is a pillar of salt.

Design:

The Philips data sheet makes design easy [tm].

Philips LM567 data sheet
FAR better for design and understanding than TI version.
Page 408 contains most of what you need to know, but there is also much more there that is useful. 



Basics:

Frequency = f0 = 1/(1.1 x R1 x C1)
   ( 2k &lt;= R1 &lt;= 20k)

Bandwidth = 1070 x (Vi/(f0 x C2))^1/2
20 mV &lt;= Vi &lt; 200 mV - input voltage

Output filter capacitor C3 - From Philips data sheet page 409
"The value of C3 is generally non-critical. C3 sets the band edge of a low-pass filter which attenuates frequencies outside the
detection band to eliminate spurious outputs. If C3 is too small,
frequencies just outside the detection band will switch the output
stage on and off at the beat frequency, or the output may pulse
on and off during the turn-on transient. If C3 is too large, turn-on
and turn-off of the output stage will be delayed until the voltage on C3 passes the
threshold voltage. (Such delay may be desirable to avoid spurious
outputs due to transient frequencies.) 
**A typical minimum value for C3 is 2 x C2

See Philips data sheet for other components.

_______________________________________

TI LM567 data sheet here - 
Detailed  design requirements are given in section 11.2.1.2 on page 14.
Shockingly poor wrt applications.    

Natsemi - looks familiar :-(.

Here is an online calculator for all tone related components

Features &amp; parameters explained they say

Lots of ideas - each image links to a page

LM567 used in IR controller



A very old but useful IC - I used these in my Masters thesis hardware 35+ years ago :-).