Do you mean using
Lion's Eye Diamond mana to pay for an
Infernal Tutor currently in your hand? If so, no, that's not possible rule-wise, as in order to get the mana from the Diamond you will need to discard your hand (including said Tutor). LED's ability can only be activated whenever you'd be able to cast an instant, and you can't cast instants in response to paying an ability or spell cost.
If you mean to use the mana to pay for a spell you search for using the Tutor then yes, simply cast
Infernal Tutor and in response to it on the stack activate LED; It won't use the stack because it's a mana ability but now your hand is gone and you have 3 mana floating;
Infernal Tutor now resolves (assuming nothing else is played in response, of course) and sees you have no cards in hand, so you get to search for anything you want and then cast it with your floating mana.