It's not going to run out any time soon - not in our lifetimes - but the environmental consequences far outweigh the benefits of peat IMO.
I used to know a scientist who would fly with the whooping cranes from canada to a brackish swamp on the west coast of Florida every year. He explained to me how peat mining was destroying their habitat and scientists were in a mad scramble researching other places they could nest other than peat bogs. There were less than 100 birds left in the wild at one point - and in the last 60 years they've barely been able to bring the population to 500 because there simply isn't enough of their habitat left to support them.
And when they drain swamps to collect the peat, they are releasing the carbon that has been building up for thousands of years, so not only is peat mining destroying more fragile, important, and irreplaceable ecosystems, it's also releasing over 1,000 megatons of CO2 per year. And that number grows EXPONENTIALLY when peat fires occur, which is sadly very common in SE Asia.
coco coir comes from coconuts, which are produced every year, and in such large quantities that it would be very difficult to drastically alter an ecosystem harvesting them. Since the nut is the only part of the whole seed used. the husk has been considered a waste item until people started amending it into their rice paddies. I personally prefer to use the pith of the husk as opposed to the coir (which is the stringy fiber) due to it's similarity to peat.