Can a jetliner use thermals to climb, or is it too heavy?

Way, way too heavy.

Airplanes are designed to meet specific mission objectives. Most airliners* are designed for efficient cruise performance. They use high-lift devices like flaps and sometimes leading edge slots to allow for slower/shorter take offs and landings. High speed cruise efficiency requires that the wings be sized to generate the required amount of lift with the lowest practical drag. That determines the size, sweep, and airfoil choice of the wings.

Sailplanes are designed to have a high lift to drag (L/D) ratio with a low sink rate. Training sailplanes commonly have L/D ratios in the 20 range while high performance competition sailplanes have L/D ratios of well over 50. L/D is also known as the glide ratio. A sailplane with an L/D ratio of 30 can glide for 30 feet (or meters) forward for every foot (or meter) of altitude lost in calm air. They're probably the most efficient aircraft anywhere. Their wings tend to be long and thin in both thickness and chord (distance from leading edge to trailing edge). Depending on the design, some sailplanes can exceed 150 MPH to race from one thermal to another and slow to a low descent rate while circling in a thermal.

Airliners are clean aerodynamically because higher drag means higher fuel consumption for a given airspeed. I've read that some airliners have L/D ratios well over 20**. However, that's at speeds far too fast to use for thermals. Besides, to take advantage of a thermal, you need to circle in it to gain altitude. To an airliner, a thermal is little more than a source of turbulence. Thermals rarely reach up to airliner cruising altitudes anyway.

*There are some special purpose airliners that are designed for short takeoff and landing performance more than for cruise efficiency. There are several excellent Canadian designs that fall into this category.

**By comparison, I doubt the L/D ratio of my Piper Cherokee light plane is more than 8 or 9. My plane just doesn't glide all that far if the throttle is cut. I sometimes wonder if Piper engineers skipped drag class.