Will the client get notified when the user invitation I sent is about to expire?
Your client will not get further notifications about the user invites other than the initial invite email. Thus, you would need to remind them when the invitation is about to expire; you can resend the invite link. This can be done from the matter you invited them to:

Frequently Asked Questions
Does the client get a reminder before their invitation expires?
No. The initial invite email is the only notification the client receives, so remind them yourself if the invitation is about to expire.
How do I resend an invitation?
Open the matter you invited the client to, go to the Client Users tab, and choose Resend Invitation Email from the Actions menu next to the invite.