There are many variables with this and I can't say for sure what you should do but consider this.

Preheat reduces hydrogen in the weld/HAZ and thus reduces the tendancy for delayed cracking. Preheat if properly applied can also reduce the residual stress at the joint. Excess preheat/heat input can reduce the strength of the material so care is needed.

PWHT is not something I am experieced with on this material so I can't say. We welded a similar material on submarines and no PWHT was performed other than controlled cooling.

I expect more will reply and give a little more information. I suggest contacting the supplier of the material and asking if they have any "data sheets" for fabrication/welding. The following is from US Steels data on how to weld T-1. T-1 is the common trade name for A-514. See the following and it may have some useful information.
http://www.ussteel.com/corp/Plate/weld/weld.htm

Hope this helps and have a nice day.

Gerald Austin
http://www.weldinginspectionsvcs.com

The material is known as a High Strength Low Alloy steel (HSLA) from the Quenched & Tempered (Q&T) type. In general this steel can be welded using any welding proces. But GTAW and GMAW are preferred because they are resulting in better impact toughness. Use of SFA 5.1 or 5.5 welding filler is common.

This type of steel can suffer from cold cracking, wich can be prevented by prehaet. But, the typical structure of A514 gained from basic heat treatment during production of the base plate material can be destroyed after excessive preheat ! High preheat results in both decrease of toughness and strength. Typical preheat for thickness 2" is 175-325 °F (105-190°C).

Another aspect is the heat input during welding and the interpas temperature : can have the same negative effect on the structure as preheat ! Welding must be done in multi-layer technique : thin layers with low heat input.

PWHT is not recommended : again negative influence on the properties.