I would start with making sure your attic has adequate ventilation. 

My professional fireplace company has two options, of which we've implemented one for our clients in the past. If you want to essentially permanently block it off, there is something like an "air pillow" which will blow up inside the flue to block it.

The solution we've used is a spring-loaded chimney cap. It sits on top of your masonry chimney flue above the roof. It has a long chain which comes into your fireplace. You use the chain to release it when you want to burn a fire. You pull the chain down to seal it when you are not having a fire.

Added advantage: better energy efficiency and indoor air quality! Keeps air from blowing out the chimney or being sucked backward through the chimney into the house!

Air sealing and insulation are the two biggest things you can do to globally improve the efficiency of the home. 

There are several articles on this application and I have yet to be in a home that didn't need it. 

Tightening up the envelope across the attic surface will lessen air infiltration at all the other areas of the home. 

It looks like the dog did a pretty good job on the door. Since the door is painted, you could sand the area down with some 150 grit sand paper. After it is sanded then you can apply some High performance wood filler or car bondo.

Apply a thin layer over the damaged area and allow it dry ( a few minutes or more depending on the amount of hardener you use). After it is dry, sand untill smooth and then apply a second layer if you need to and then sand smooth. After the area is the way you want it, prime it and paint it. 

Hello Alex, 

One alternative that most homeowners are not aware yet is reviewing your Roofing System. Along with the attic insulation, the type of roof you have can make a huge difference on how energy efficient your home is.

Most roofing materials (especially asphalt composite roofs) absorb solar energy and transfer heat to your home (requiring more air-conditioning during summer). On the other hand, energy efficient Metal Roofs are reflective and emissive. They bounce most of the sun's visible and UV light - meaning less heat transferred to your home. Independent studies show energy savings of up to 25%.

The Interlock Metal Roofing System is Energy Star certified (US only). More than energy efficient, our roofs are truly sustainable, being made of up to 95% recycled material, reducing the dependence of asphalt and stopping the cycle of old roofing material going to landfills. It is a lifelong solution that is better for your home, your pocket and the planet.

The most cost effective things that you can do to improve your homes energy efficiency will depend on the existing condition of your building envelope, the efficiency of your equipment, and humiditiy control.

Addressing the condition overhead and tightening the building envelope are basic principles involved in maintaining efficient temperture control in the home. Thermal imaging will clearly indicate areas of major heat loss.

There are many types of structures and styles of thermal barriers. Therefore, you are best advised to have an energy consultant from a Home Energy Rating Service HERS evaluate of your particular home and assist in identifying the weakness of your home envelope and machanical equipment. Such a report  can provide you with a roadmap to begin taking steps to improve the overall efficiency of the home. 

Verify that all window and door seals/weatherstriping is in good shape.

Check to be sure doors and windows are caulked properly (no gaps, cracks or missing).

If you have an attic access, treat it like door opening and seal properly. If possible, create an insulated cover to go over it in the attic.

Check HVAC ductwork for leaks and seal with foil tape or mastic.

Wrap your hot water heater in a blanket.

All of the answers given were great options. There are many, many ways to improve a home's energy efficiency. As we are a siding and window company, my focus will be on those areas. Windows are a huge energy loss/gain. Did you know that windows and doors account for approximately 50% of your heating/cooling loss? Air leaks around a window or door due to a crack of 1/16" of an inch is like having a hole in your wall the size of a brick. In the average home, that is equal to about 15 bricks. Consider how large that "hole" is in your home. The glass package is the most important thing to consider when replacing windows. Most folks just look at the cost of the window and install, and some smoke and mirrors some companies like to drag on about. A smart homeowner knows what Low-E is, what a good U-Factor rating is, whether or not it's single, double or triple pane glass, etc. etc. Same features apply to most doors. Lastly, siding is another area where a homeowner can save money. Check into insulated siding. Check to see what kind of underlayment is being applied to your home. Many companies don't even put a housewrap on to save cost and the average homeowner never knows. Make sure you only hire someone who is appropriately licensed and insured. Ask to see those documents. Hope this helps folks looking to replace siding, windows or doors! Happy remodeling!

If you do not know what you are doing, hire someone who does!