MSN has this one article that keeps getting recirculated on it's own feed. (http://www.msn.com/en-ca/foodanddrink/foodnews/foods-you-should-never-reheat-and-why/ss-AAlrn95?li=AAggFp5&ocid=mailsignout#image=1)
First off MSN is a horrible source. It often shares nonsense articles about nutrition with very little to almost no scientific references or credibility. The original sources can usually be traced to quackery sites like Mercola and Natural News which spew an abundance on absurdities aimed at the gullible. This "article", or rather clickbait, gives very little substance and resorts simply to the fear-mongering. As usual with stuff like this, there is often a small grain of truth, but important information is missing in order for a consumer to make an informed choice. Most of these become problematic if reheated from something that was left standing at room temperature (and therefore allowing bacteria to do their nasty work). If proper preparation and storage (fridge/freezer) procedures are followed, there will be little to be concerned about.
1. Celery,beets, spinach, lettuce: Yes these food can contain nitrates. Nitrates can be converted to nitrosamines. It is thought though that other substances, like vitamin C, within the plants help protect against them. To help prevent/reduce bacterial and enzymes refridgerate any unused food immediately.
2. Potatoes and Mushrooms: Yes foods heated do lose some nutrient value, but the amount is small. The benefits of heating foot outweigh the small loss and include easier digestion (and thus more nutrient absorption), decreasing of bacteria, and taste enhancement. Really no reason not to reheat if following proper food production and storage (don't let unused food stand too long unprotected and at room temperature allowing for bacterial growth).
3. Chicken and eggs: Just like the other items above, the only real problem is with contamination if left out and not in the fridge. The big concern is salmonella poisoning if proper storage and handling protocols are not followed.
4. Rice: Unique fact, reheating properly stored rice (again to prevent bacterial growth) is actually healthier as it has a lower carbohydrate amount (the same is true of pasta as well)
In regards to the oils, I don't have knowledge in that right now, but I would at least have to agree that they don't taste the best reheated, so just for that reason go with fresh oil every time.