Whether you should keep your left heel on the floor during your backswing is really a question of flexibility.
The initial answer to this question is yes - you should work on keeping your left heel down during your backswing as this will allow you to achieve maximum power during your swing.
You create power during your backswing and therefore distance in your shots, by creating the most possible twist, or coil, between your upper and lower body. You want to achieve a 90 degree turn to the right (for right handed golfers) in your upper body and shoulders. However, maximum power is achieved by creating the greatest possible difference between how much your shoulders turn and how much your hips turn. Ideally, if you can keep your left foot down and on the ground and rotate your shoulders 90 degrees to the right, you will create power. By only rotating your hips 40-45 degrees to the right, you will achieve a 45-50 degree difference between your hips and shoulders, a great deal of coil or twist in your torso and therefore you will achieve maximum power. However, if by keeping your right heel down you are unable to rotate your shoulders to close to 90 degrees right, this will start to affect the angle that you swing the club on, or your swing plane. By not rotating your upper body or shoulders correctly during your golf swing, you will now find it very difficult to swing the club head correctly and approach the golf ball along your target line. The club head will move far enough away from the target, on the inside, to allow you to approach the ball on the downswing from the inside of the target line. Not turning your shoulders well on your backswing will result in a more over the top movement, whereby the club head will move forwards of you and result in the club head cutting across the target line as you hit the ball. This will result in shots that either fly straight left, swing left or slice, depending on the club face position at impact.
If you are unable to rotate your shoulders 90 degrees right whilst keeping your left heel down, then allow your left heel to rise slightly. This will allow your hips to rotate slightly more than 40-45 degrees right but with this slight increase in hip turn, you will achieve the 90 degree rotation in your shoulders and will find it easier to approach the ball correctly with the club head on your downswing. Only raise your left heel the minimum amount needed to allow your shoulders to rotate correctly and make sure you maintain your head height, rather than allowing your head to move upwards as your heel moves up.
To improve the flexibility between your shoulders and hips you could try following a golf flexibility improvement programme but check with your doctor prior to doing this.