Is it okay not to take insulin injections before sports or any other physical activity?

The answer to your question is somewhat complicated and depends partially on what kind of insulin regimen you are using ( two shots a day of fast and intermediate acting, vs fast acting before meals along with some kind of longer acting insulin, or an insulin pump). What you need to know to figure out the answer in your particular case is the following:

In most, but not all individuals, exercise tends to lower the blood sugar either during, immediately following, or sometimes hours after the exercise. Most people need to either eat more food and/or take less insulin working during exercise.
Exercise will only lower the blood sugar if some insulin is still working in the body (either insulin injected soon before the exercise with a meal or injected hours earlier, but still working during exercise (or very rarely, if the body is still making a lot of insulin on its own during the honeymoon or remission phase).
If you exercise when there is not enough insulin working at the time, your blood sugar will almost certainly go up and you may spill ketones and this could be dangerous.
Very rarely in some people, the blood sugar will go up after exercise even when they take their usual dose of insulin.
If you are taking a just small amount of fast-acting insulin with food before exercise, but there is still a significant amount of insulin working in the body anyway (perhaps a longer acting insulin taken earlier), you may be able to skip the insulin before exercise. More likely, you will just need to lower the dose.

You need to work with your own doctor to figure out whether you need to lower your insulin before, during or after exercise and/or eat extra food. It is usually best to start with a small amount of exercise and see when and how your blood sugars change and then gradually increase you exercise and adjust you insulin and food as necessary working with your own doctor.

TGL