To elaborate and expand a bit on the older answer kindly linked by u/Annual-Impression377...

Yes, Greeks routinely climbed Olympus. But before outlining why they did that, we should probably begin with a more basic question: did the Greeks really believe that their gods lived on Olympus?

The answer is more complicated than you might think.

Olympus was always both mountain and metaphor. In the Iliad, for example, Olympus is at once an actual peak (with epithets like "snowy" and "craggy") and a metonym for the heavens. At one point, for example, Zeus boasts:

"If you tied a chain of gold to the sky, and all of you, gods and goddesses, took hold, you could not drag Zeus the High Counselor to earth with all your efforts. But if I determined to pull with a will, I could haul up land and sea, then loop the chain round a peak of Olympus, and leave them dangling in space. By that much am I greater than gods and men." (Iliad 8.19-26)

This Olympus is obviously something more than just a snow-capped mountain in Thessaly. One gets the same impression from the myth of Bellerophon, who sought to fly Pegasus to Olympus and join the gods. (Zeus was not pleased by this act of hubris, and saw to it that the hero never reached the summit.)

The dual conception of Olympus as both physical peak and heavenly realm continues throughout Greek (and later Latin) literature. The discrepancies between these conceptions are clear in the mythological compendium known as Apollodorus' Bibliotheca (probably written in the second century CE). In a myth about twin giants who attempted to storm the homes of the gods, the author notes:

"When they were nine years old and measured eighteen feet across by fifty four feet tall, they decided to fight the gods. So they set Mount Ossa on top of Mount Olympus, and then placed Mount Pelion on top of Ossa, threatening by means of these mountains to climb up to the sky" (1.53)

There's a clear differentiation here between Mount Olympus and the home of the gods. The distinction in even clearer in Lucian's Icaromennipus, a satirical second-century text about a man who decides to fly to the home of the gods. Mennipus (the protagonist) doesn't bother with Olympus; he sets sail directly into the sky, and figures that the gods live very far off indeed. To quote his calculations:

"Let me see, now. First stage, Earth to Moon, 350 miles. Second stage, up to the Sun, 500 leagues. Then the third, to the actual Heaven and Zeus's citadel, might be put at a day's journey for an eagle in light marching order..."

By the time Lucian wrote, the Greco-Roman elite had various theories about the true home of the gods. Each philosophical sect had ideas on the subject. The Stoics claimed that the divine principle was implicit in all things. The Platonists (and, to some degree, popular belief) scattered the gods among the stars. The Epicureans assigned them a blissful realm far from the troubled world of mankind, or denied their physical existence altogether. In this context, calling the home of the gods "Olympus" was purely conventional - nothing more or less than an expression of the ubiquitous influence of the poets and the myths.

But this doesn't mean that Olympus was regarded as just another mountain. Olympus is a massive - well - massif, with many sub-peaks and several prominences that appear to be nearly the same height. We don't know whether the Greeks ever climbed to Mytikas, the highest summit. We do know, however, that they routinely ascended the sub-peak now called Hagios Antonios, a short distance from (and only slightly lower than) Mytikas. There, from the third century BCE to the fifth century CE, offerings were made at an altar of Zeus.

The Greeks may have thought that this was the summit. Even if they knew that it was slightly lower, they could see the other peaks from this vantage point, and it would have been obvious to anyone who sacrificed at Zeus' altar that there were no gilded palaces or sunbathing deities on the peak. There seems, however, to have been a lingering sense that Olympus was a special place. According to Solinus, writing in the third century CE:

"The things that are to be seen at Olympus show that Homer did not celebrate it rashly. First, it rises so high, with a preeminent peak, that the inhabitants call the top of it heaven. On the summit is an altar dedicated to Zeus. If burned offerings of entrails are brought to it, they are neither blown off by windy breath nor washed away by rain, but as the year rolls on, whatever is left there is discovered unchanged; what is consecrated to the god triumphs over time and the corruption of the air. Letters written in the ashes remain until the next year’s ceremony." (8.6)

In the post-classical Greek imagination, the gods may not have lived on Olympus. But nor were they necessarily far away.

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I made a short video on this topic last year, which is basically just an abridged version of this answer. Decent pictures, though.