I’m paranoid about the high temperature contact with the plastic and ending up smoking something toxic after relighting later.
Get a glass or metal ashtray. The worry is not worth it.
For tobacco cigarettes it’s safer than the cigarette … put it out and fire it up later shouldn’t be a problem … or any more of a problem … just look, chunks of plastic dangling off the end and maybe just pitch it …
Smoke the whole thing, problem solved. 
Where’s that damn “like” button …
When I was only smoking a half a cigarette at a time, my common practice was to simply knock the cherry off, then blow through the cigarette (to make sure it’s out as well as getting all the soon to be stale smoke out of it). When you do it that way, and you get pretty good at totally breaking off the lit part, it really doesn’t matter what you do it on. Hell, I could put one out on wet concrete and still be able to relight it later.
When done this way, there’s no cross contamination between the cigarette and whatever you’re putting it out on since you’re just using that to break off the lit part.
I keep a pair of scissors next to my ashtray and snip off the end of the joint or cigarette. It wastes a little but you get a clean relight.
Curiously, (way) back in the day I noticed that joints would not stay lit, unlike regular cigarettes. Today, by contrast, that no longer seems to be true; I have no idea why that is, unless it’s the greater availability of cannabis generally, and in particular of pre-rolled joints in the states where it has been legalized (under state law)*.
*Qualification intended to acknowledge the discrepancy between federal law and the laws of certain states which are at odds with it.
I dimly understand the technical prowess of those posting in this thread who aren’t me…
but there are so many perfectly good glass ashtrays, big, small, plain, fancy, languishing in so many places - why not just grab one and have nothing to worry about?
Put the joint out with your tongue, making sure to take a drink right beforehand.
/pro-tips
I assume joints don’t stay lit (as well) because they tend to not be rolled well. That pot just isn’t packed in well enough to sustain the lit end, especially if someone isn’t puffing on it. All it takes is a gap in there somewhere, bug enough that the embers can’t bridge it.
Also, [some] cigarettes nowadays will go out, by design, if not puffed on for more than a set amount of time (30 seconds? a minute?). This is so they stop burning if you forget about them and they’re less likely to fall out of an ashtray and start a fire.
Most smokers know that if you leave a cigarette alone, it’ll fall out of the ashtray once it burns past the part in contact with the edge of the tray.
Whatever “toxins” from the ashtray the OP might encounter are probably less of a concern than pesticide residues in the joint itself.
And with the current hysteria over minute amounts of glyphosate residues and an unproven cancer link, you can bet that growers will be turning to far more dangerous herbicides instead.
The ashtray would be the least of my worries.
In the 70s/80s, my pot was seed filled and drier than California. I used a pair of manicure scissors to roughly cut it up. Joints were rolled by hand. Pieces of stem ended up in the joint.
Fast forward to today. My weed is seedless and dripping resin. I use a grinder to get a uniform product that I roll in a sweet little rolling machine.
Totally different things.
Heh, this is like asking if while playing Russian roulette you should worry about carpal tunnel from spinning the barrel. 
Although I guess it’s the classic, I ordered two Big Macs, a fry and a McFlurry with a Diet Coke trope. Just because you’re doing unhealthy things doesn’t mean you want to compound it with more.
So, in that vein. it all depends on the plastic. Many plastics are carcinogenic when burned. The question then is simply is the ashtray going to melt or not. A good one maybe with a polyimide coating will not and the risk is essentially negligible. LDPE though? I might worry about it a bit more. PVC is a death trap, but I doubt you would find many people making ashtrays out of it, but I’d stay away from Wish when buying my ashtrays. Polypropylene I would say is the iffy case (and I would say it’s also the most likely thing to make an ashtray from.) It could melt from a cigarette, but it might not ignite. Here’s a great chart with ignition temps of all sorts of materials. http://www.tcforensic.com.au/docs/article10.html
You made me LOL with the Russian roulette analogy. Good one.
I got something like this. It’s supposedly some heat-resistant sort of plastic.
https://www.walmart.com/ip/Mainstays-4-Plastic-Ashtray-In-Assorted-Colors/167593003
…and right afterwards.