No easy way. Grab an edge and start pulling. Use a spading fork to loosen it as you go. I imagine that this isn’t what you want to hear, and i hope that it isn’t covering the whole yard, but if it is there, I know of no way to remove it other than by hand, and your sod won’t be successful if it is left there. Perhaps you can hire some eager 12 year olds who would love some extra spending money.

I would try a sod cutter if you know where all the sprinkler heads are. If the fabric is very old it often tears apart when you pull on it. I don't envy you.

This is no single kind of weed barrier/landscape "fabric," and what kind you have would make a difference. There is the cheap, thin, plastic type of "fabric" usually sold retail. It is not woven but extruded as a thin membrane. It is not durable and rips apart in bits as you try to pull it up. Then there is the woven polypropylene that is more like a trampoline mat. It is durable and can resist tearing during the removal process. If you have the former, I think it would take several lifetimes to actually remove. Since it is non-functional as a durable barrier, I'd ignore it except for what pieces might be exposed at the grade surface. Get rid of them for cosmetic reasons. We already that plants can grow on top of and through it, so I don't think it would cause a lawn serious harm. Also, it's possible to dig through it in order to install other landscape trees and shrubs.

If, on the other hand, you have the much more durable woven polypropylene fabric, it's also going to be much more in the way of future landscape projects ... minor grading, planting and general plant maintenance. I would try to remove it but think it would be a difficult job without machinery. I've never had to remove it so am guessing about the process, but think that a rented Bobcat would be the machine most likely to have success, because it can scrape off a layer of earth, and it can pull with serious muscle. That's my 2-cents on it.

I wouldn't use a sod cutter for removal if the fabric is a woven plastic material.......unless you have it on its lowest cutting setting so it is below the fabric layer. It will get caught and tangled up in the equipment. Encountered this issue personally with the nylon netting often used to grow sod. Had to stop and clean out the tanged mess every few feet.

A bobcat or skid steer would be a much better and faster method.