I suppose Voldemort couldn't use the Patronus Charm. So, what would happen if he faces Dementors? How can he counter them?
He could use a way other than a Patronus.
Though we don’t know what method Snape preferred, Snape disagreed with Harry on the best way to fight Dementors. Therefore, other ways of fighting Dementors exist.
“Harry fully expected to receive low marks on his, because he had disagreed with Snape on the best way to tackle Dementors, but he did not care: Slughorn’s memory was the most important thing to him now.”
- Harry Potter and the Half-Blood Prince, Chapter 21 (The Unknowable Room)
There’s at least one other way to fight the effects of Dementors seen. Sirius was able to stay sane while in Azkaban because they were unable to suck the thought of his innocence, which wasn’t a happy thought, but an obsession.
“I don’t know how I did it,’ he said slowly. ‘I think the only reason I never lost my mind is that I knew I was innocent. That wasn’t a happy thought, so the Dementors couldn’t suck it out of me … but it kept me sane and knowing who I am … helped me keep my powers … so when it all became … too much … I could transform in my cell … become a dog. Dementors can’t see, you know …”
- Harry Potter and the Prisoner of Azkaban, Chapter 19 (The Servant of Lord Voldemort)
Therefore, the Dark Lord would be able to use another method.
He usually got them to be allies, though.
Though it’s likely he could with a method other than a Patronus, the Dark Lord didn’t typically have a need to fight Dementors. As he tells the Death Eaters, he knows the Dementors will join them.
“The Dementors will join us … they are our natural allies … we will recall the banished giants … I shall have all my devoted servants returned to me, and an army of creatures whom all fear …”
- Harry Potter and the Goblet of Fire, Chapter 33 (The Death Eaters)
He was able to turn them to his side since he was able to offer them more than the Ministry.
“The rest of us sleep less soundly in our beds, Cornelius, knowing that you have put Lord Voldemort’s most dangerous supporters in the care of creatures who will join him the instant he asks them!’ said Dumbledore. ‘They will not remain loyal to you, Fudge! Voldemort can offer them much more scope for their powers and their pleasures than you can! With the Dementors behind him, and his old supporters returned to him, you will be hard pressed to stop him regaining the sort of power he had thirteen years ago!”
- Harry Potter and the Goblet of Fire, Chapter 36 (The Parting of the Ways)
The Dark Lord was proven right when he did get the Dementors to join him, and they aided in his second rise to power.
“I thought Dementors guard the prisoners in Azkaban?’ he said cautiously.
‘They did,’ said Fudge wearily. ‘But not any more. They’ve deserted the prison and joined He Who Must Not Be Named.”
- Harry Potter and the Half-Blood Prince, Chapter 1 (The Other Minister)
It’s possible that the Dark Lord might have needed to fight Dementors if he had nothing to offer them, but since he did, he likely wouldn’t have needed to.
- While the Patronus charm is the most widely known method of combating dementors, it is not the only one. Sirius Black was able to resist their effects while in Azkaban by focusing obsessively on his own innocence and periodically taking the form of an animal, whose less complex thoughts could not be affected as severely. Other magic which affects the mind, such as occlumency (which Voldemort did know how to use), could also theoretically be used to resist the effects of a dementor attack. When teaching Defense Against the Dark Arts, Snape specifically marked Harry down for writing that the Patronus was the most effective way to fight them (though this should be taken with a grain of salt as Snape was prone to marking Harry down for whatever he felt like).
- Dementors target the soul of the creatures that they face, which in the case of Voldemort was so fractured from his horcrux experiments that they may not have been able to fully affect him to the same degree. The "dementor's kiss" pulls the soul from the body, but Voldemort's soul was mostly contained in other objects. This may also have inured him somewhat to the effects of their happiness-draining powers.
- Voldemort claimed that dark wizards and dementors were natural allies, as they will join with anyone willing to give them a constant supply of souls to feed on. This was shown to be at least partially true when the dementors joined Voldemort's side in the second Wizarding War.