The animals are described to be incorporeal, but seemingly can still touch dementors (which aren't).
Given that the above is true, can you touch a Patronus?
During the hearing in The Order of the Phoenix Harry is questioned by Madam Bones
"You produced a fully fledged Patronus?"
"Yes," said Harry, "because -"
"A corporeal Patronus?"
"A - what?" said Harry.
"Your Patronus had a clearly defined form? I mean to say, it was more than just vapor or smoke?"
The definition of corporeal:
- of the nature of the physical body; bodily.
- material; tangible: corporeal property.
So, Yes, you should be able to touch a fully corporeal Patronus.
There is, perhaps, evidence that a patronus lacks a physical body that can be touched. In Prisoner of Azkaban when Harry casts the patronus that saves their past selves, we have the following description:
It stopped on the bank. Its hooves made no mark on the soft ground as it stared at Harry with its large, silver eyes.
That its hooves made no mark implies that it is not physical. Similarly, immediately before this we have another description:
It was cantering back toward Harry across the still surface of the water.
This too might indicate the patronus's lack of physicality. Unless there is additional magic that allows a patronus to levitate, if it had mass it would make a mark on the surface of the water (and probably sink as well). We also find other patronuses that apparently move through the air:
And then a silver hare, a boar, and a a fox soared past Harry, Ron, and Hermione's heads:
Perhaps even more explicit, though, is the description of Snape's doe in Deathly Hallows:
It was a silver-white doe, moon-bright and dazzling, picking her way over the ground, still silent, and leaving no hoofprints in the fine powdering of snow.
Snow crunched beneath his feet but the doe made no noise as she passed through the trees, for she was nothing but light.
Here again we see that the patronus had no physical impact on its surroundings, and even more importantly it is specifically stated that it is just light.
Notably, the one time that Harry does try to touch his patronus, back in Prisoner of Azkaban, it disappears:
But as his trembling fingertips stretched toward the creature, it vanished.
On the other hand, we do find a patronus apparently physically interacting with dementors. From the dementor attack in the beginning of Order of the Phoenix (my emphasis):
An enormous silver stag erupted from the tip of Harry’s wand; its antlers caught the dementor in the place where the heart should have been; it was thrown backward, weightless as darkness, and as the stag charged, the dementor swooped away, batlike and defeated.
"THIS WAY!" Harry shouted at the stag. Wheeling around, he sprinted down the alleyway, holding the lit wand aloft. “DUDLEY? DUDLEY!"
He had run barely a dozen steps when he reached them: Dudley was curled on the ground, his arms clamped over his face; a second dementor was crouching low over him, gripping his wrists in its slimy hands, prizing them slowly, almost lovingly apart, lowering its hooded head toward Dudley’s face as though about to kiss him...
"GET IT!" Harry bellowed, and with a rushing, roaring sound, the silver stag he had conjured came galloping back past him. The dementor’s eyeless face was barely an inch from Dudley’s when the silver antlers caught it; the thing was thrown up into the air and, like its fellow, it soared away and was absorbed into the darkness. The stag cantered to the end of the alleyway and dissolved into silver mist.
(It is hard to argue that the patronus and dementor could interact because both are not physical, since it seems pretty clear that the dementor physically interacted with Dudley. Yet the dementor is also described as "weightless as darkness", which could imply that it is not really physical.)
According to "Fantastic Beasts and Where to Find Them", the answer is a resounding yes, under the correct circumstance, a corporeal Patronus can interact physically.
In his description of how he was able to evade a lethifold, Flavius Belby offers us this account.
Pointing my wand away from myself into the deadly folds of the creature, summoning the memory of the day I had been voted President of the local Gobstones Club, I performed the Patronus Charm.
Almost at once I felt fresh air upon my face. I looked up to see that deathly shadow being thrown into the air upon the horns of my Patronus. It flew across the room and slithered swiftly out of sight.
Notably, although the Lethifold seems to share some common features with the Dementor, it's classed in the book as a "beast", indicating that it has a physical presence.