In High Fantasy settings it always seems like centuries and sometimes millennia go by with no technological advancement. Is there any reason why?
To build on top of what Daniel said, there are really three core concepts behind the stagnation of technology in high fantasy settings. Not all apply to Arda but they apply in different measures in different fantasy worlds.
Most of human history, as Daniel pointed out, takes place in periods of stagnated development. Sometimes the result of the downfall of a civilization, and the loss of technology. Sometimes due to religious reasons. Sometimes just due to being so early in the development of a species that advancements have little to build on.
Progress, despite what we are taught about genius and hard work, is more a function of all the required pieces being in place to enable the new advancement. Today, we live in a time of unbelievable advancement. However, the advances from the Bronze age to the Iron age took some 2000-5000 years depending on where we are talking about.
The idea of lost technology plays a role in the different ages within LotR. However, it is mostly about lost forging techniques and would be analogous to the loss of being able to make Wootz or Damascus steel. This is why a first age sword might be so highly prized in a LotR context. This is exactly like how a ~2000 year old Wootz blade might cut through modern armor in the year 1600.
Not a big force in LotR, but present somewhat. Progress is always about doing something better than the prior process. If the prior process is a magical one that makes a 200 Ton stone door move like it were made of cardboard, it is very hard or impossible to create a mechanical process that can compete. And it would be even more unlikely that anyone would bother to try. The presence of magic almost ensures a slowed-down progression of technology for this reason alone.
Additionally, there is a brain-drain associated with the magical arts. While not really a factor in LotR lore due to the rare nature of true magicians. In most other high fantasy worlds the mentally gifted enter magical schools. They learn magical arts, not engineering. They learn to create potions, they learn 'mend wounds' spells, not how to suture a wound with thread. All the smarties are advancing magical science, not physical science.
Progress really isn't stagnated at all. Science and technology are about learning to understand the world as it is, and learning to make use of those facts. If your world is like ours and lacks demons, spells, enchantments, magical teleportation, dragons, so on; then your technology and science will bear that out and you will advance along the same threads that we have.
But if your world features those things, and at the strength that most fantasy worlds give those powers, advancement will rightfully be along those lines instead.
In many fantasy settings, magic attempts to look like it did in our own past. This means extreme ritualism. This is a form of anti-science and does not seek progress. It seeks tradition. I once heard someone say, of modern magic and science, that "Science is the attempt to eliminate from rituals that which is unneeded, and magic is the attempt to honor the tradition of the rituals that produced the refined technology."
Steel-making was once, and in some places still is, a ritualistic and magical process. One common way to produce steel involved folding the iron, binding a slip of magical paper to the block of iron, pounding the paper into the iron, folding again and repeating 100 times. When we strip away the layers, you are adding carbon to iron which is exactly what is needed to make steel. The magic worked. The ritual was passed down from Master to Apprentice and without any knowledge of the underlying reason, steel was producible.
This is how Wootz steel could be created ~3000 years before we had any idea what carbon nanotubes were.
A society that has given themselves over to ritual, would by definition not progress. Many fantasy worlds are dominated by ritual, and yet can't be said to be in a dark age, because those rituals produce really advanced results. There are historical examples of this. Ancient Egypt was able to build some of the most enduring landmarks, create a nuanced and long-lived culture, and be very productive and even culturally progressive all while being very ritualistic. They did so by adopting the rituals of others along the way. However, if the whole planet were Egypt, then they probably would mirror many of these fantasy worlds. And they would have retained their rituals much longer, and they would have been more powerful, as in the fantasy world the gods would be real.
The term is "high fantasy".
There's two responses to this. Firstly, I don't think it's necessarily true. Especially in Tolkien, we see all sorts of different levels of technology - the Hobbits, for example, seem to be a just pre-industrial society: they have water mills, but nothing more complex. Saruman, by contrast, is clearly gearing up Isengard into an industrial age. But other civilizations have different "technologies": the Rings and the Silmarils, for example, are both examples of technology in that world, although we wouldn't recognise them as such. They're works of craft, definitely.
The other point to make is that long periods of technological stasis are the norm, even in the real world. Our current break-neck speed of technological innovation is very much atypical in history. Not much changed in the western world, for example, between the time of the Romans and the invention of printing.
For most of human history, technology was almost without any progress. Humans took several tens of thousands of years to invent such basic technologies as the wheel, metallurgy, and similar techniques. It took several thousands of years for the neolithic "revolution" to spread throughout the inhabited world, and hundreds of years for the industrial revolution to do the same. Computers took several decades, the Internet (and mobile communication) one decade to take over the world.
The speed increases exponentially. No matter which part of human history you look at, significant technological progress only ever happens at the end of that part. Living in a time of increasingly fast technological progress, we tend to forget that most of our history almost nothing of technological interest happened.
Necessity is the mother of invention.
Using the Middle Earth examples... The Hobbits lead generally simple lives and valued artisanal products--ale, tobacco, etc. They enjoyed working in their gardens. Granted, they didn't have factory-produced goods for comparison, but still... Nor were they curious or at war with anyone. Their needs were met, so there was no desire or drive for anything "better."
Some authors of fantasy fiction do specifically mention periodic disasters which have depopulated the lands of their fantasy setting and would certainly set back any technological progress.
For example, Tolkien writes of wars with Wainriders and the Witch King during the 2nd and 3rd Ages and the Great Plague of the Third Age which severely depopulated Middle Earth. Tharbad was abandoned after a flood. Smaug destroyed the Dwarves's Kingdom Under the Mountain.
In the lands of Westeros from George R. R. Martin's Song of Ice and Fire books, the erratic seasons have caused periods of prolonged winter for many years (presumably resulting in mass starvation and a return to a hunter-gathering existance in the north).
These are civilization destroying calamities. Low intensity wars and plagues that reduce the population by 10% or 25% may have no long term consequences, and could actually spur on technological development. But if 90% of the population is wiped out every millenium, I imagine you'll constantly be hitting the reset button for innovation and progress.