In my view Corribus has provided 2 excellent responses both worthy of mole snacks!
I don't have a lot of substance to add, but I can say that in this case an increased degree of polymerisation is not responsible for a decrease in flammability. Depending on which source you choose to accept, Cotton has a typical degree of polymerisation somewhere between 3000 and about 15,000. By contrast Viscose has an average DP of about 250-450, so the primary influence on flammability is unlikely to be DP. Another key structural difference in these cellulose based polymers is average degree of crystallinity. Cotton (at least un-mercerised cotton - primarily Cellulose I) has a much higher average crystallinity which lies somewhere between about 60 and 70%. Viscose (based on the Cellulose II allomorph) has a much lower average crystallinity which probably lies somewhere between about 40 and 60% (depending on the results/methods of analysis you choose to accept). Could this be responsible for the difference in flammability? Probably not.
If we are discussing 'naked' fibres, then personally I would back Corribus' view that the differences flammability are down to physical rather than chemical differences. Cotton is on average a thinner fibre (0.7-2.5 D) than Viscose (1.5-15 D) and I would imagine that its c shaped cross section would expose much more of its surface area, leading to more rapid burn. Viscose tends to have a crenelated cross section which although of greater surface area than say a plain round fibre, is still probably of lower surface area than a typical cotton fibre.
If we are discussing fabric rather than individual fibres, then a lot of this will be down to weave and fabric weight. Your nice new Levis are made of cotton fibres, but they are heavyweight and twill woven (a very closed weave), leading to low flammability. By contrast your nice new cotton T shirt consists of about 1/2-1/3 the number of fibres per unit area than your jeans and is probably plain weave or brushed cotton with a much more open structure meaning that its a lot more flammable.
Anyhow... I shall shut up now!