Tim offered some great suggestions. In addition to following some of his advice, I might suggest some other possible solutions looking at the problem from more of a mechanics, muscle stress view.
First of all, if you are playing in classical position without a footstool, and you play typical right hand guitar, your guitar would rest on your right knee (in classical position) so you should be tapping your left foot (the one your guitar is NOT resting on). It's not just the weight of the guitar but the position you must hold your leg in to keep the guitar balanced on your leg that might place additional stress on the muscles used to tap your toe.
Even if you are tapping the opposite foot from the one attached to the leg which supports the guitar, the seating position can play a big role in increasing or minimizing the fatigue induced by the movement of raising your toe or heel.
You will find it much less strenuous to lift your toe if your foot is placed out in front of your knee. If your heel is directly under your knee or your leg is curled such that your heel is behind your knee (which might be the case if your guitar is on the foot tap leg), too much stress will be placed on the muscles used to raise your toe. Experiment and you will see what I mean.
Also try altering your seating height which will change the geometry of your lower leg as it relates to the floor. You will find that some seating heights (too high or too low) will place additional stress on the muscles used to raise your foot. Experiment to find a balance between a comfortable position for your guitar and a position that minimizes the stress on the muscles used to lift your toe. You might find that wearing a strap even while seated will permit you to vary your seating position without relying solely on your leg to support the guitar.
Be sure your are low enough that your heel is firmly on the ground so that you can use your heel as a fulcrum. It may also help some to try different type shoes to see if shoes with higher heels make it less or more stressful to lift your toe from your preferred seating position.
If you do prefer a seating position that places your toe under your knee, then you might find it easier to raise your heel rather than your toe. Some positions may even allow for heel and toe to be raised alternately - as suggested in Tim's great answer.
Other methods of keeping time might include nodding your head to the beat or a rhythmic movement of your torso (like dancing while playing). Remember, you don't need exaggerated movements, just a steady, continuous, evenly spaced movement that you can detect.
Keep experimenting with different positions and different ways to keep the beat while you play and eventually you will settle into something that is comfortable and effective for you personally - that won't induce unnecessary muscle stress or fatigue. Good luck!