Here is a great article from the University of Michigan that covers some of the topics relevant to your question.
The article deals with the Net Primary Production (NPP) of CO2 in the world related to human consumption. NPP is the amount of CO2 that is "fixed" (i.e., processed) by plants through photosynthesis minus the amount of CO2 that is produced by organisms through respiration. Therefore, to simplify things, the higher the NPP the lower the amount of CO2 in the atmosphere.
The article cites the following table from Atjay et al. 1979 and De Vooys 1979 which breaks down NPP by the type of ecosystem:
(The numbers in the table are amount of surface area on Earth in km2 x 106 and the second number is the NPP in petagrams.)
- Forest: 31 / 48.7
- Woodland, grassland, and savanna: 37 / 52.1
- Deserts: 30 / 3.1
- Arctic-alpine: 25 / 2.1
- Cultivated land: 16 / 15.0
- Human area: 2 / 0.4
- Other terrestrial (chapparral, bogs, swamps, marshes): 6 / 10.7
- Lakes and streams: 2 / 0.8
- Marine: 361 / 91.6
The article uses these figures to calculate the amount of NPP that is currently co-opted for human consumption. For example, they assume that all of the NPP associated with cultivated land goes toward human consumption. They conclude that 30.7% of the terrestrial NPP and only 2.2% of the aquatic NPP is co-opted by humans. These numbers are of course based off of ~30 year old studies, but I think it's still safe to conclude that there's still room for more human-based CO2 production before NPP goes to zero.
According to this 2002 study by Randerson, et al., terrestrial heterotrophs (i.e., organisms that need to breathe oxygen, like humans) produce 82–95% of the CO2 represented by the NPP. Let's be conservative and assume the higher amount: 95% NPP. That means that as long as the forests account for fewer than 5% of the total NPP then we should be fine. The forests produce 48.7 Pg, however, which is a little under 22% of the total NPP.
But wait! 13.6 Pg of NPP associated with forests is co-opted for human consumption (e.g., getting wood for building houses, &c.). If we were to get rid of all of the forests we'd also get rid of that percentage that has already been co-opted. Therefore, if we were to get rid of all of the forests there would be a net loss of 48.7 Pg - 13.6 Pg = 35.1 Pg, which is about 16% of total NPP. That's just low enough to meet the lower bound of 82% CO2 production.
Therefore, there is a small chance that there will be enough NPP after getting rid of all of the trees for human consumption, but it is likely not the case. Furthermore, if we were to get rid of all NPP producers other than grass there would certainly not be enough NPP for human survival.
It's also important to note that, by far, the most productive producers of NPP are the open ocean, tropical rainforest, and temperate forest (see Figure 5 of the University of Michigan reference), so by deforesting as opposed to de-grassing we would be greatly reducing the efficiency of the global ecosystem. Also, as I mentioned in a comment above, there is also the matter of carbon storage. Trees store a good amount of the carbon from the CO2 they process in their trunks where it stays for a long time. Grass, on the other hand, releases its carbon back into the system shortly after it dies and rots away. Therefore, even if grass does produce enough oxygen for life, it probably wouldn't have the same greenhouse gas reducing capabilities as trees.