TPS61240 Boost converter IC is not working at all?
I see two problems.


You say you are using a 1206 type inductor in your comments, but the pad connected to pin 5 on your PCB is HUGE, at least for an inductor 4 or maybe 6 times as large as that. Unless that's not an inductor, but a smoothing cap, which would be a problem in itself, because that's where the inductor should be.
At first I was sure you weren't placing/connecting things right, but now I'm thinking you are connecting pin 4 and 6 (EN and Vin) in a rather round-about way through the big component that's either wrong or your inductor. If the Vin at least has a cap close, that may be okay, but it's not the neatest. If there's no cap near Vin with simple tracing for the power in, it'll probably choke on the trace inductance or resistance or some such.


Which leads me to wonder: Are you sure you placed the pins in the package in the right order?

If you did and you got confused about the inductor you are using: Solder a decent cap directly onto the Vin pin with a strong connection to the ground trace. With a booster that needs only 1uH you need to be very wary of any trace inductances and resistances. Also maybe beef up the ground trace (and Vin trace) with some solder or copper wire.