Can a Nikon D3200 camera shoot HD infrared videos?
Entirely impossible. Non specialized devices cannot shoot anything meaningful in the infrared range, even if you remove the IR filter.
DSLR sensor can grab at max 70-75% of the so called "Near Infrared", which accounts for just the 0.15% of the entire infrared range, so a DSLR without the IR filter is capturing what you could consider IR noises, more or less.
Conclusion, if you are dreaming (as many do) to take real infrared videos or photos...forget it. To put thing into perspective, two of the best IR cameras out there -the FLIR E60 and the Fluke Ti400- have a resolution of 320x240 and cost nearly 8000 dollars, far far away from a used DSLR camera with its IR filter removed.
On the other hand, if you want to add something "different" to your photos and videos, as small as it can be the IR noises range you can capture with your modded camera can work quite well.
The Nikon D3200 (like most other DSLRs) has an IR filter on its sensor. You have two ways of making it do IR photography and both involve some DIY work and have their caveats.
- You could use an IR transmitting filter (example link) which tries to clip out most of the visible and UV light but lets in IR light. With this filter you will need long exposures to collect sufficient photographic detail (because the IR filter on the sensor will cut out most of the IR light). There is another issue with this scheme -- most of the optics you use are also designed to clean up stray wavelengths (IR/UV), hence longer exposures. Further, the sensor's filtering mechanism might create internal reflections that add up as noise in your long-exposure capture. All together not the best performing option as far as IR photography goes -- but, your camera can continue to be used for its optimized visual light work.
- You could take out the sensor IR cut-off filter (here is a popularly available tutorial, they will also do it for you at a price). At this point you have lost IR protection for your regular (visible light) photography. So, to do that you would need to use IR/UV blocking filters. But there are drawbacks to this method too (as described in this local answer).
There is a third and better option for your IR photography. Get a second older digital camera and remove its IR filters (method two above). You use this for IR photography now and keep the D3200 safe.
Why this third approach?
You will lose your visual light photography quality when you temper with the IR filter (this is my subjective opinion, others may have a different opinion). With IR filters (option 1) you lose on the IR photography quality. If you are biased to IR photography, option two might work, else getting a second-hand DSLR and tweaking it for IR would be a good path. If the D3200 is your second DSLR and you have another primary DSLR, you are already on this path
Another interesting article to browse,
How to turn your DSLR into a full spectrum ‘super camera’
ExtremeTech: David Cardinal on January 1, 2013