There is a tendency to think that fires will increase in the near future due to global warming. This is because many fire risk prediction are based on climate only. However fire regime changes not only depend on climate ; there are other factors, like land-use changes, CO2, plant invasion, fragmentation, etc. that are also important drivers of change in fire activity . Even plant drought stress (and flammability) not only depends on climate [2,3].
A recent simulation study  suggests that global burned area is certainly predicted to increase in the following decades when simulations are based on climate only (blue line in the figure below). However, adding the effect increased CO2 reduces the predicted burned area to no increase (green line below). Furthermore, when adding increased population density and urbanization (black and red lines), the model predicts much more area burnt in the last century (black lines 1900-2000) and a reduction of future burned area (red lines). The predicted reduction of fire during 1900-2000 is consistent with global charcoal records  and can be explained by increasing agriculture, land use and fragmentation. Overall, this study suggests that global area burned is unlikely to increase in the following decades.
Note that 1) this is a model, so take it with caution! 2) This model is at the global scale, but changes in different directions are expected in different regions, and this can have biodiversity consequences (even if the global balance is steady); for instance, in the Mediterranean Basin, fire are likely to keep increasing as land abandonment and fuels are increasing . And 3) there is a high uncertainty in some fire drivers. For instance, temperature is likely to keep increasing, however, rainfall and wind changes are very uncertain, and landuse and emissions are subject to uncertain changes in environmental policies in different countries. In any case, this study gives us an idea of the possible sensitivity of different parameters.
Figure: Simulation of global area burned for 1900 to 2100 under different scenarios: a) climate only (blue line); b) climate + CO2 (green); c) climate + CO2 + population & urbanization (black lines; red area for the future predictions). From .
 De Cáceres M, et al. 2015. Coupling a water balance model with forest inventory data to predict drought stress: the role of forest structural changes vs. climate changes. Agr. For. Meteorol. 213: 77–90. [doi | pdf | suppl. | blog]
 Luo, Y. & H. Y. H. Chen. 2015. Climate change-associated tree mortality increases without decreasing water availability. Ecol, Let. 18:1207-1215.
 Knorr W, Arneth A, & Jiang L, 2016. Demographic controls of future global fire risk. Nature Clim. Change 6:781-785.
 Marlon JR, et al. (2008). Climate and human influences on global biomass burning over the past two millennia. Nature Geosci, 1, 697-702.