He means they are crap people buy because they're cheap (used to be anyway) and then when they get some money they buy something better.

I think he may be referring to "disposable" cars
Not sure I agree
But, If your going to road trip comfort is key
I do a lot of road trips and the best one for that was a chevy suburban (04)
Over 400 miles on a tank
Very comfortable seats 
The worst was a tie between a 16 crosstrek and an 05 altima
The seating is torturous in the best of conditions
The crosstrek was gutless on the interstates while the altima held its own
my .02c

Burner cars can have two really different definitions.

One definition is disposable like others have suggested.  Like a burner phone. Something cheap and easily replaceable.

The other definition is a heavily modified ( Japanese ) car to look and/or be faster than stock.  Rice burner or rice rocket is a synonym for this term.

While the Yaris, matrix, and, fit are cheaper, I personally don’t see them as disposable.  All of 5ese are fine vehicles.

my mechanic says they are "Burner" cars. Would you be able to elaborate on this or give your opinion?

As I understand it it, he says these are "Disposable cheap junk that you don't buy to keep for a long time"

And that's mostly true they're just meh.

The only one of these that isn't "that bad" is the newer American Yaris models are actually quite good, if you're alright with it's 'driving characteristics'.

It's just a rebadged Mazda2, which is known for very poor breaking performance, so test it to see if the car you're looking at breaks reasonably quickly.

Additionally, these cheap 1.5L engines have issues with injector noise (very quick tick-tick-tick-tick...) and I wouldn't advise buying one if the noise is so loud you can hear wether the car is running from 10 feet (direct injection injectors are VERY expensive to replace).

And lastly safety, it's decent - but I'm not sure how it'll hold up in a collision with a heavy truck.

So yeah, it's a mixed bet - usually people don't buy these to keep for decades, mostly "what's the cheapest and most reliable way to commute"

I can't think of a good small car currently sold in the US, so quite honestly I'd just get a third generation Mazda3 (2014-2018) or a late production tenth generation Toyota Corolla (2011-2013) 

I also want to do roadtrips and I would rather have a gas saver than a comfortable suv

If you're shopping new and are looking for a regular gas car, well you can't beat the RAV4. additionally the Toyota Corolla Cross and Toyota C-HR are all great cars. The Mazda CX-30 and Mazda CX-5 are honorable mentions, very well built cars.

But if you want the new car to also be good on gas, although the Kia Niro isn't the best built car - it's inexpensive and it is ridiculously good on gas in my experience and the warranty coverage of the hybrid drivetrain is also a big bonus over their competitors. 

Shopping for a used Crossover / SUV is though, There's a lot of demand and prices are incredibly high - in the current state of the market I would not buy a used SUV.

Besides that, out of all the cars I've drove - I must say that the most fun you can have on a road trip is with a small coupe. Driving a large bulky SUV gets boring after a while.

He probably meant that they are cheap piece of sh** cars because they are very small, ride like sh**, uncomfortable, and are slow as can be. I personally wouldn’t buy any of those because of the small size and rough rides. Even corollas and civics are too small for me. I’d rather be in a 20 year old Camry than a brand new Yaris. But that’s just personal preference. Pretty reliable though