Coilgun - IR sensors interference
I'm not going to go into any math, just some general principles.


You need decoupling capacitors on your LM393.  These are (relatively) small capacitors that you install right at the IC power pins.  Usually like 100nF.  Very  short leads.  Like, millimeters not centimeters.
Your sensors and LEDs are on long(ish) wires.  The wires will pickup any pulses that occur when you switch off a coil.  Use twisted pairs for your sensor wires, or twisted pairs with a grounded shield.  This will minimize the intereference that the wires pick up.
You use a potentiometer to set the trigger threshhold for your sensors.  You need a capacitor from the potentiometer wiper to ground.  In your example, that's pin 2 of the potentiometer.  Again, something like 100nF.  This will keep noise and intereference from disturbing your threshhold setting.
Coils store a lot of energy.  When you switch them off, that energy has to go somewhere.  When you just switch one off, the energy has to go somewhere.  A lot of it goes into an electromagnetic pulse.  This induces currents (and voltages) in nearby conductors - including your circuits.  You need shielding (enclose your circuits in metal boxes) and you need to reduce the intensity of the turn off pulses.  Instead of just opening the circuit to the coil, switch it to short the coil through a (low value) resistor.  That will absorb the energy instead of transmitting it.  It also changes the time it takes for the magnetic field to die, so you may need to experiment with the discharge resistor (optimize the time) or modify the timing between the coils.
Another thing to consider is how you run your wires.  The orientation of the wires to the coils influences the amount of interference the wires pick up. You probably can't get rid of all of the interference this way, but if the wires are run just absolutely the wrong way it can make a bad situation much worse.


That's all just general stuff.  Really, it all comes down to minimizing the interference your coils generate and making your circuits "immune" to the intereference that you can't avoid.



You've added the coil driver circuit to your question.

That brings me to another suggestion:

It looks to me like you have the drivers centralized, with long wires to the coils.

I think you ought to put the driver for each coil right at the coil.  

The wires between the driver and the coil will act like antennas.  The longer they are, the more interference they will transmit.

Shorter wires, less interference.

Run the 80V power and the control signals to each coil, with your switching circuit right there at the coil.

Use short wires, and twist them together.

Use the thickest wire you can to distribute the 80V to the drivers.  Each driver would also need a large(ish) capacitor on the 80V connection (on the driver board) to make up for higher resistance and inductance on the power supply.

Use twisted or shielded wires on your control signals going to the drivers.