Should these be placed in refrigerator: tomatoes, and bananas, just like green leaf vegetables?
You generally do not want to refrigerate tomatoes; they will will get mealy in texture, which can be unpleasant.
Bananas are more complicated. Once you refrigerate them, ripening will stop, and it will not recommence when they are brought back to room temperature. So you never want to refrigerate unripe bananas.
Ripe bananas can be refrigerated to make them last a few days longer, but the skins may darken, which some people find unacceptable.
Tomatoes loose their sweet flavor when refrigerated. Ideally, to have the best result they should be kept in cool, basement like temperature. Something between 55 to 70 degree F is a very pleasant one for tomatoes.
Unfortunately that kind of a condition is not available in most households, however, if flavor is important to you, invest in a very small refrigerator($90.00 ones), so you can keep its temperature on cool (65F) and store tomatoes and maybe other fruits in that refrigerator.
For Bananas: Ideally, do not refrigerate them, but if you have plenty, here is how to save and properly refrigerate them.
First, leave them at room temperature until they are just barely ripe, so if you've bought them green, they would be barley ripe on the third day. You want to see no signs of greenness, and a light yellow color.
Next, separate the bananas, and individually wrap each one with a newspaper (tightly rather than loosely). Place them in a plastic back and store in the lower drawer of your refrigerator.
After this, they will last up to 7-10 days. Take them out one or two at a time and costume quickly.
Fruit should never be stored in common domestic fridges (4°C). And tomatoes are a fruit (fruit is anything with seeds in it)
The exception of course is if you have a modern refrigerator with proper climate control, e.g. Fisher & Paykel Cool Drawer running in pantry mode
Most fruits undergo a slow enzyme ripening process (part of their natural life cycle), this is usually interrupted at low temperatures (<10°C) which leads to strange textures and tastes. Also most refrigerators are effectively dehydrators which will damage the outer layers of the fruit
If you have too much fruit on hand that is going to expire, you can also just freeze it whole for a future sauce, jam, or baking project