I do.
Against champions such as LB or Swain, keeping track of their mana bar is literally a matter of life and death. In the top lane, many champions have OP mechanics such as Hecarim's helicopter swings and E knock back or Jax's counter strike. However, they can only use them when they have the mana to afford them, and often times, these abilities cost a ton of mana. (1/4 of their pool early on.)
The opponent's mana bar is probably one of the key factors in deciding when to move in for a kill when they are low. Fighting aggressively against an Irelia or Olaf while they still have mana is probably suicide in most cases.
In the bottom lane, most supports have sure-kill abilities that costs 1/3 of their mana pool before level 6. Deciding to play safe and farm or take the initiative and poke against a Blitzcrank or Leona is also hugely influenced by their mana bars.
On the flip side of the coin, I would advise against fixating too much on the enemy's resource. Mana is a fluid mechanic, and there are really no ways to gauge precisely how much mana someone has quickly. Being too over confident that an opponent can't fight back because they are OOM might just spell death for you as they suddenly gain mana back through regeneration or leveling up.