No. your free account is independent from your paid plan and you can keep both accounts separately.
QR Codes and Landing pages from a free account do not expire.
Take into account that they have a limitation of 100 scans/day that you cannot exceed or the account will get blocked.
You can import QR Codes from your free account to a paid plan.
However, it is not possible to transfer QR Codes from a paid account to a free account.