Thermostat Replacement?

Picture below is for 2.5L engine. Drain coolant down, remove hose from elbow, remove bolts from elbow.

No. Physically the engine will run but it won't get up to normal operating temperature where parts expand and fit properly. Excessive engine wear will take place, there will be more "blowby" than normal causing sludge in the oil, and that blowby will be less likely to vaporize and be drawn out and burned. The Engine Computer won't go into "closed loop" which is when it looks at the readings from the oxygen sensors to fine tune fuel metering. That results in excessive fuel consumption and more blowby.

If you're thinking about removing the thermostat to solve a problem, it might change the symptoms but it isn't going to fix anything.

REMOVAL
Drain cooling system to the thermostat level or below.
Remove coolant recovery system (CRS) hose and thermostat/engine outlet connector bolts Thermostat/Engine Outlet Connector - 2.0L Engine and Thermostat and Engine Outlet Connector - 2.4L Engine 2.0/2.4L Engines. 2.5L engine remove inlet hose and coolant elbow from thermostat housing Thermostat, Housing and Inlet Elbow - 2.5L Engine
Remove thermostat assembly, and clean sealing surfaces

Thermostat is located where the uper radiator hose goes into the engine usually, that is where it is.