Though your cell phone requires a plan of some sort to operate most of its functions, your pictures and other data are not reliant upon having service. Your phone stores certain data on its internal storage, whether that be its internal memory or an SD card if you have one installed. Unless you delete these pictures from your phone, they will last until you decide to transfer them or reset your old phone to factory settings.
Since your phone doesn't have service, you can't use your phone's data plan to transfer your pictures to another device. If you have Wi-Fi in your home, however, your phone can still connect to that and you can email your photos to yourself or use a cloud-based app. Alternatively, if you can remove your phone's SD card and have the right adapter, you can directly transfer your pictures from your SD card to your computer. You may first need to transfer your pictures from internal storage to your SD card if they are not stored there by default.
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