Feasibility of Design: Using a TLC555 to Oscillate Three IR LEDs at 37kHz (schematic, datasheet, and calculations included)
Your resistor values are much too low. Use a much smaller capacitor and much higher resistor values. You have 0.25\$\Omega\$ for the Rt so the TLC555 will attempt to sink 20 amperes from the 5V supply. Not good. Absolute maximum is 150mA and you really don't want to be anywhere near that high.
You can go down as far as about 100pF for the timing capacitor and still have predictable timing, lower still if you don't mind tweaking.
You need a series resistor on the transistor base. The 15mA is a limit you must ensure, not a current the chip will limit it to.
Note: You should have some idea of what accuracy you are shooting for. You may not get within 10% of the predicted center frequency if you use wildly low or high components. Here is what I think is similar to you are trying to interface with (a Vishay part), but you should use the actual datasheet for the device you are using:

You really should be within 1% or so of the nominal center frequency to maximize the sensitivity.