Dell Optiplex 790 - Can it Run Minecraft?

Minecraft requires Java and a minimum Radeon HD6450 at which case it will run but not on high resolution.

https://help.mojang.com/customer/portal/articles/325948-minecraft-system-requirements

Recommended Requirements
Pentium 4 3.0GHz
Athlon XP 2600+

GeForce 8600 GS
Radeon HD 6450

2Gigs system Ram

When people on a budget ask me I tell them to look for a Dell Precision 380 with 4 gigs of ram and add a Radeon 6450 GPU.

http://www.ebay.com/sch/i.html?_nkw=dell+precision+380

Very doubtful using the onboard Intel graphics. You need at least the low profile AMD Radeon HD6350 video card added to that system. The Optiplex line were not designed for gaming. Look at the XPS line of desktop computers.

Try the demo.

https://help.mojang.com/customer/portal/articles/425381-minecraft-demo-mode

My kids have been playing minecraft for years on a variety of systems.

The low-end was a Dell Inspiron 17R Core i3-2310M with 4GB of memory and integrated graphics.

It does OK but can occasionally get laggy.

An Inspiron 15 with a Core i5 2410M and integrated graphics is working well.

My point is minecraft is not very graphics intensive compared to other current games. 

edit:

Maybe I mis-understand. If you are looking to buy that system to play minecraft, I'd look for a little more system, and lean towards newer architecture not older, and one that can easily take a graphics card upgrade.

I personally don't prefer the small form factor systems as they are harder to upgrade.

What budget do you have overall? That will drive a lot of this.

My max budget is $550

Here are a couple of build your own.

PCPartPicker part list: pcpartpicker.com/.../tDGHLk
Price breakdown by merchant: pcpartpicker.com/.../by_merchant

CPU: Intel Core i3-4160 3.6GHz Dual-Core Processor ($111.99 @ SuperBiiz)
Motherboard: MSI H81M-P33 Micro ATX LGA1150 Motherboard ($44.85 @ OutletPC)
Memory: Crucial Ballistix Sport 8GB (2 x 4GB) DDR3-1600 Memory ($58.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($47.89 @ OutletPC)
Video Card: XFX Radeon R9 270 2GB Core Edition Video Card ($147.98 @ Newegg)
Case: Fractal Design Core 1000 USB 3.0 MicroATX Mid Tower Case ($34.99 @ SuperBiiz)
Power Supply: EVGA 500W 80+ Bronze Certified ATX Power Supply ($37.99 @ NCIX US)
Optical Drive: Samsung SH-224DB/RSBS DVD/CD Writer ($15.88 @ OutletPC)
Operating System: Microsoft Windows 8.1 (OEM) (64-bit) ($87.75 @ OutletPC)
Total: $588.31
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-03-06 00:57 EST-0500

Here is a little less money: only a dual core CPU. Radeon 260X. cheaper but perhaps too big of a case, less costly and quality PSU.

PCPartPicker part list: pcpartpicker.com/.../hxXFyc
Price breakdown by merchant: pcpartpicker.com/.../by_merchant

CPU: Intel Pentium G3258 3.2GHz Dual-Core Processor ($71.98 @ Newegg)
Motherboard: MSI H81M-P33 Micro ATX LGA1150 Motherboard ($44.85 @ OutletPC)
Memory: Crucial Ballistix Sport 8GB (2 x 4GB) DDR3-1600 Memory ($58.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($47.89 @ OutletPC)
Video Card: MSI Radeon R7 260X 2GB Video Card ($118.99 @ SuperBiiz)
Case: Thermaltake VL80001W2Z ATX Mid Tower Case ($22.99 @ Micro Center)
Power Supply: Corsair Builder 430W 80+ Bronze Certified ATX Power Supply ($19.99 @ Newegg)
Optical Drive: Samsung SH-224DB/RSBS DVD/CD Writer ($15.88 @ OutletPC)
Operating System: Microsoft Windows 8.1 (OEM) (64-bit) ($87.75 @ OutletPC)
Total: $482.31
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-03-06 01:10 EST-0500

from Dell outlet this one is $359.00 

Add a $40 PSU and a $150 graphics card and you are right at $550.

Video Card: XFX Radeon R9 270 2GB Core Edition Video Card ($147.98 @ Newegg) 
Power Supply: EVGA 500W 80+ Bronze Certified ATX Power Supply ($37.99 @ NCIX US)

Caveats and excuses, that graphics card may not fit. It was to use as an example.

Here are a couple of build your own, with prices for comparison.

a dual core..

CPU: Intel Pentium G3258 3.2GHz Dual-Core Processor ($71.98 @ Newegg)
Motherboard: MSI H81M-P33 Micro ATX LGA1150 Motherboard ($44.85 @ OutletPC)
Memory: Crucial Ballistix Sport 8GB (2 x 4GB) DDR3-1600 Memory ($58.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($47.89 @ OutletPC)
Video Card: XFX Radeon R9 270 2GB Core Edition Video Card ($147.98 @ Newegg) Power Supply: EVGA 500W 80+ Bronze Certified ATX Power Supply ($37.99 @ NCIX US)
Optical Drive: Samsung SH-224DB/RSBS DVD/CD Writer ($15.88 @ OutletPC)
Operating System: Microsoft Windows 8.1 (OEM) (64-bit) ($87.75 @ OutletPC)
Total: $529.30

I'd invest a little more for a core i3 vs the dual core G3258

CPU: Intel Core i3-4160 3.6GHz Dual-Core Processor ($111.99 @ SuperBiiz)
Motherboard: MSI H81M-P33 Micro ATX LGA1150 Motherboard ($44.85 @ OutletPC)
Memory: Crucial Ballistix Sport 8GB (2 x 4GB) DDR3-1600 Memory ($58.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($47.89 @ OutletPC)
Video Card: XFX Radeon R9 270 2GB Core Edition Video Card ($147.98 @ Newegg)
Case: Thermaltake VL80001W2Z ATX Mid Tower Case ($22.99 @ Micro Center)
Power Supply: EVGA 500W 80+ Bronze Certified ATX Power Supply ($37.99 @ NCIX US)
Optical Drive: Samsung SH-224DB/RSBS DVD/CD Writer ($15.88 @ OutletPC)
Operating System: Microsoft Windows 8.1 (OEM) (64-bit) ($87.75 @ OutletPC)
Total: $576.31

Prices include shipping, taxes, and discounts when available