Will the PC build support 2 or 3 monitors for gaming?

Solution
You have a high end GPU in that build so you should be able to expect good gaming across multiple monitors (within reason of course). Your issue will be with the CPU... It will need a good overclock to assist in driving your high resolution and frame rate. Add a CPU cooler to that build (Hyper 212 EVO), overclock to 4.5GHz and that would be the max potential you could hope for without going with an FX 8000 series CPU. By comparison, the Intel build below would manage your gaming goals a bit better without overclocking... If you can stretch the budget to upgrade to the i5-4460 you would be in very good shape. I also swapped the case out for the 200R.

PCPartPicker part list / Price breakdown by merchant

CPU:...
You have a high end GPU in that build so you should be able to expect good gaming across multiple monitors (within reason of course). Your issue will be with the CPU... It will need a good overclock to assist in driving your high resolution and frame rate. Add a CPU cooler to that build (Hyper 212 EVO), overclock to 4.5GHz and that would be the max potential you could hope for without going with an FX 8000 series CPU. By comparison, the Intel build below would manage your gaming goals a bit better without overclocking... If you can stretch the budget to upgrade to the i5-4460 you would be in very good shape. I also swapped the case out for the 200R.

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i3-4150 3.5GHz Dual-Core Processor ($119.99 @ Amazon)
Motherboard: ASRock H97M PRO4 Micro ATX LGA1150 Motherboard ($67.98 @ Newegg)
Memory: G.Skill Ares Series 8GB (2 x 4GB) DDR3-1600 Memory ($78.99 @ Newegg)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($54.99 @ Amazon)
Video Card: Sapphire Radeon R9 290 4GB Tri-X Video Card ($249.99 @ Newegg)
Case: Corsair 200R ATX Mid Tower Case ($39.99 @ Newegg)
Power Supply: XFX 650W 80+ Bronze Certified ATX Power Supply ($49.99 @ NCIX US)
Total: $661.92
 
Solution
Your GPU will support as many monitors as the number of ports it has. R9 290 has 4 ports and so will support 4 monitors, provided you have the cables/adaptors for all monitors to be connected to that graphics card.
I use my r9 290 with 3 monitors currently and has no issues adding a TV on as a 4th monitor