My furnace is located in a closet next to the hot water heater. The area is closed off by 2 bi-fold slotted closet doors. This area happens to be near the living room so when the furnace runs it gets quite loud.

I was thinking that using solid doors instead of slotted should help with the noise. However I'm unsure if that would be an issue with heat and/or ventilation of the area.

Does anyone have some advice? I'm open to any other noise reduction tips as well.

Do not use a solid door as both the heater and the furnace needs air to breathe!

If you make provisions for the furnace and water heater to get fresh air you can do what you described.

If your house is not a two story house you could run a 8 inch pipe up into the atic and down to the level of the burners on the furnace, Then do the same with the water heater with a 6 inch pipe.

I`ve seen instances where the installer planned for the slotted bi-fold doors to serve as the return air,, do you have a return air system coming in from somewhere else,, some people do wild and weird things sometimes

Well there are 2 pipes going to the furnace, but I'm not sure where the ends are. I've included a picture so you can see my setup.

Picasa Web Albums - Ben

Also I noticed that there is a draft coming in from around the ducts as they feed up into the rest of the house. Is it safe to stuff some insulation around them?

The two pipes you have now are not for combustion air, One is the furnace flue and the other is a discharge heat pipe. If you are going to put in solid doors you have to add combustion air as discussed.

Combustion Air

Calculator

House of Craig - Online Combustion Air Calculator

Providing Enough Combustion Air

One of the pipes in your picture is for the exaust fumes to leave the building through the roff. I think the other pipe is for or was for a humidifier.