No. The return your child earns on both Cash Junior ISAs and Investment Junior ISAs are tax-free.
For an Investment Junior ISA the fund your child's plan invests in grows free of income tax and capital gains tax (other than tax on dividends from UK shares).
For a Cash Junior ISA there's no tax to pay on any interest earned on your child's savings.
Tax treatment depends on individual circumstances and tax law may change in the future.