Hmmm, lots of interesting information and ideas
Cotton is a plant fibres (cellulose) and that makes it awkward to dye using natural materials. It's not that you can't dye it, just not all that easily.
Firstly, is it coated with anything? A waterproofing or fireproofing agent ? 'Cos if it is, you'll have bother getting anything to dye it until that's been removed. Usually for canvas, soap and water and a big scrubbing brush will do. At the size you are talking that's a lot of elbow grease though.
For dyeing I think Eric's idea of smoking it is probably the easiest, though Dave is right too and some will wear off. Don't use birch unless you like tarry cloth.
Cellulose fibres need to be tanned before they are mordanted and then re-tanned and then dyed to do an effective job. The vital bit there is tanned, so any of the tannin rich substancess, be that tea or coffee or tree bark or galls or nuts will work on their own. But, you need not only the dye stuff made into a liquid, but you need a vessel big enough to move your tarp around in to let the dye take evenly.
Might I suggest that you make your own camouflaged dyed tarp ? That way even ness of dye is not important and you could make up various natural dyes and paint them on having first mixed them in a thickener, lilke gelatine or wallpaper or flour paste. It will be incredibly messy, but once the gunk is washed off you will have a coloured tarp. Whether it's the colour you want
ah well, now that's up for debate
However, it will overcome both the need to soak the entire tarp at once *and* any problems about getting the tarp soaked through......which technically you ought not want the cloth to be able to do anyway.
Generally the natural dyes that will work best for this are tea, madder root, walnuts, or some of the imported bark dyes. They need to be tannin rich and in themselves substantive dyes that need no mordants.
Best of luck with it, let us know how you get on ?
cheers,
Toddy