All people release glucose from the liver when they're fasting. The amount varies.
There are some people with T2 who have lot of insulin resistance and may release 3 times more glucose that a non diabetic would, so when they fast their levels can rise considerably.
The amount released also varies in people with T1. It also depends on insulin resistance, which can vary from person to person (age and weight play a part.). If someone with T1 fasted and took no basal insulin, then their blood glucose levels would rise. In a few cases, it can take only a few hours without basal insulin for a person to develop very high glucose levels, without any insulin they may also develop high ketone levels.
If someone uses a basal insulin and manages to adjust their a basal insulin correctly; then in theory, during short fasts, then their blood glucose should remain the same or vary by just a mmol/l or so. However, the liver releases varying amounts of glucose throughout the day and night so this perfection is often very difficult to achieve with just one or two injections of basal insulin.
People on pumps can adjust their basal insulin on an hour by hour basis. They can programme their pump so that enough insulin is released to counter the predicted release of glucose from the liver. So for example they might need a higher rate in the early morning to combat dawn phenomena and they might need less when they are active. If they've got the adjustments right then fasting for a while would be fine (indeed they use fasting test to determine whether the pump is set up correctly)
However, fasting for a long time would be a different 'ball game'
If someone was using mixed insulin then they would not be able to fast since the rapid part of that insulin is used to cover mealtime needs and without the normal amount of carbs then they would have a hypo.