No, flight hours do not expire. As long as your flight hours are listed in your logbook and signed off by an instructor then you do not need to worry about losing the hours of experience that you have gathered. Flight hours are incredibly important when it comes to getting jobs in the aviation industry, so it is important that you always get your hours signed off by an instructor to make them valid.
That being said, while flight hours do not expire, there are some time limits you might encounter when getting flight experience. Some regulatory qualifications and requirements mean that you need to complete a certain amount of hours within a fixed period of time. These hours do not expire, but if you fail to complete the required amount of hours within the fixed time then this could impact whether they can count towards your active flight hours.
But generally speaking, no, flight hours do not expire. Every time that you pilot a plane or helicopter then you are actively gaining experience, in terms of hours, and these will last for your whole life.