Being responsible for a job or chore can be a win-win for parent and child. Offering a job or project as an opportunity rather than giving it as an unending assignment is what keeps it growing in the future.
For your child, learning to be responsible for something requires learning how and when to say “yes,” and if it is not a personal choice, then it is not an opportunity to be personally responsible. The process of agreeing to be responsible for a particular job or project, whether invited to or not, is the second part of the learning process.
For a parent, this means 1) giving your child the option to decline your request, as in an invitation, and 2) being sure the agreement has clear conditions, including a beginning and an ending time. When the ending time is not obvious, put a time or clear condition on it.
Finally, let your child know (don’t assume) it is safe to ask you for support and advice. If you child wants to quit, be understanding. What has changed? Does he or she feel appreciated? Acknowledge that sometimes we adults agree to do something and discover that it doesn't work for us. Trust yourself to look from your child's point of view, and choose whether allowing your child to quit (without a harsh relational emotional penalty), or be understanding but firm about keeping the commitment. If you are not sure what would be best, cancel the agreement. If your child is ready for it and it is set up well, your child will not be reluctant to take it on again or to take on more. Allowing an occasional withdrawal of a promise is a good exercise in learning to be responsible for others' expectations. This maximizes the learning as well as the likelihood that your child will accept more responsibility later.
[There is a little known axiom of human behavior that is well-known in physics: force causes resistance. Sometimes force is necessary, by the way, but even then it causes resistance. You choose whether it is worth it, and then share those thoughts - including that being or feeling made to do something frequently can make you not want to do it.]