Short answer: Not really.
Long answer: It depends.

But let’s take a deeper look. Nitrogen in tyres is used frequently in industrial and aerospace applications, in fact, regulations stipulate aircraft tyres must contain no more than 5% oxygen due to issues with brake system overheating and causing tyre explosions . Nitrogen is used due to the fact that it’s non-flammable, easy to store, easier to manage than requiring an air compressor/air drying equipment and it's easy to handle and transport.

Lastly, unless a leak is severe enough to drastically displace the available oxygen in the breathable atmosphere, it’s completely non-toxic in normal amounts. 

Nitrogen makes up 78% of the air that surrounds us. The remainder is 21% Oxygen, 0.9% Argon and the rest is made up of trace gasses like Carbon Dioxide, Xenon, Neon and so on. As you can see, the overwhelming majority of air is already Nitrogen.

What does this have to do with tyres?...we did get a little sidetracked there.

What are the myths about Nitrogen filling tyres?
In Australia, a red valve cap indicates the tyre has been inflated with nitrogen gas rather than air

In Australia, a red valve cap indicates the tyre has been inflated with nitrogen gas rather than air

You won’t need to check your tyre pressures.
“Filling your tyres with nitrogen will dramatically reduce the natural pressure loss that occurs when normal air permeates through the rubber that makes up your tyres.”

Technically, this is true. O2 will permeate through tyre rubber about 3-4 times faster than N2 due to oxygen having a smaller molecule size than nitrogen. Although the size difference is only 2.6%. 

Since 78% of air is already nitrogen, we can say that 78% of that myth is bogus.

That being said, filling your tyres with nitrogen does NOT mean you can skip checking your tyre pressures. You will still suffer pressure loss to a varying degree (lots of factors involved), and pressures ought to be checked between fortnightly or monthly. With gauges being so cheap, you really don’t have any excuse. Checking your tyre pressures regularly really is the cheapest and easiest way to get the most from your tyres.

Nitrogen reduces internal oxidising in the tyre.
Come on now, nobody has ever replaced a tyre because the internal surfaces were oxidised. While it is true that oxygen is corrosive, in the time that it would take to cause any real damage to the tyre, the tyre would have been worn out and (hopefully) recycled decades prior.

Besides, even if tyre oxidation were a real concern, we’d see oxygen affecting the deep layers on the outside of the tyre too, regardless of what the tyre was inflated with.

Nitrogen responds better to temperature changes.
False. Both Oxygen and Nitrogen are ‘ideal gasses’, which means that their pressure and temperature rise at roughly the same rate. We found a good article here about how the pressure of different gasses change with temperature (spoiler alert: it doesn't make any difference).

The best thing about nitrogen filled tyres is the lack of moisture. Typically, if you inflate your tyres with a simple compressor or hand held pump you will introduce a small amount of moisture into the tyre. As moisture reacts to temperature changes at a higher rate than air you could possibly see higher pressures on air than with ‘dry’ nitrogen filled tyres. You could see similar benefits using a compressor with an industrial-grade air dryer, which is typically out of the price range of the average workshop or home garage.
Checking tyre pressure with a gauge

It's easy and quick to check your tyre pressures with a cheap gauge.

So what’s your point?
Basically. it boils down to this: For the everyday commuter or consumer, filling your tyres with nitrogen is something that has no drawbacks but also negligible benefits in the real world. All the benefits an average consumer will get from Nitrogen filled tyres can be gained for free by regularly checking and maintaining correct air pressures. The added benefit of checking pressures frequently is catching leaks and punctures early.

If you can get it for free? Go for it! Your lap times will thank you. Maybe.

If you're asked to pay for it? Don't worry about it, go buy yourself a doughnut or four instead.

If you already have a vehicle that has nitrogen filled tyres, don’t worry, you can top them up with regular old air without any negative side effects.

Final word: If you’re eking out the last seconds from your lap times and performance is key, it might be worth considering Nitrogen filled tyres. Reducing the moisture inside your tyres will result in more consistent temperatures and pressures, otherwise tell ‘em they’re dreamin’.