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Most hard cheeses that you would buy at a cheese shop or at the cheese counter do not require that you store them in the refrigerator. However, many soft or semi-soft cheeses, particularly shredded or cream cheeses must be refrigerated for safety reasons.
Many people do not refrigerate their hard cheeses, and although that is totally fine, cheese that is refrigerated will last longer. For example, a block of unopened hard cheese can be good for up to 6 months in the fridge. So, if you plan on cutting and eating your block of cheddar that weekend or over the next few days, you really don't need to make space for it in the refrigerator.
A food-safety note: the USDA says perishable foods shouldn't sit out for more than 2 hours (1 hour if it's above 90°F / 32°C), and that your fridge should be 40°F (4°C) or below. For cheese, USDA guidance treats hard cheeses like cheddar, Swiss, Parmesan and provolone as the forgiving ones, which fits with leaving them out for a while. Soft cheeses, fresh cheeses and shredded cheeses are a different story: keep those in the fridge, and don't leave them out for more than a couple of hours.
Considerations
Other things to consider before you stop refrigerating your cheese:
Temperature control
So let's say you're not going to put your hard cheese in the refrigerator, that might be okay. Many people make the argument that hardened or mould ripened cheeses are not stored in refrigerators--but they are kept in temperature controlled rooms--that temperature is generally 50-55° F (10-13° C). The typical refrigerator is 35-38° F (2-3° C), so a refrigerator is considerably cooler than a cheese cave, but your house is likely not that cold either.
Contamination
Just because you're not putting your cheese in the fridge, doesn't mean you don't have to worry about contamination. Outside of the fridge you have good and bad mold spores, germs, insects and other household pests--yes, mice do eat cheese.
To prevent contamination when storing your cheese on the counter, make sure to wrap it like you would if you were putting it in the fridge; either wrapped in paper or better yet wax cloth (Amazon). Keep the cheese in a closed container or invest in a cheese dome (Amazon) to help keep the cheese in and the germs and pests out.
Advice for keeping cheese in the fridge
If you decide to keep your cheese in the fridge, that's totally fine also. You might want to consider keeping your cheese in the vegetable drawer--not only is the vegetable drawer the warmest area in your refrigerator, but it also keeps at a consistent temperature. Every time you open your fridge you're letting cool air out, keeping the cheese at a consistent temperature keeps your cheese fresher longer.
Related: How should I store leftover cheese? · Can cheese be frozen? · How can you tell if cheese has gone bad? · Can you cut the mold off cheese? · How long does cheese last?