If you have a general journal entry that is an infusion of capital of say 10,000 cash, I would debit cash and credit capital for the amount.

However, when you post the entries, where do those balances go? What happens to the general journal entries? Do they stay for the entire fiscal year or are they removed at the beginning of each period?

Cash is an asset so it will be carried forward to the next year. Similarly capital is the equity and will be carried forward to the next year. However, in some accounting package, one needs to make a formal book entry to carry forward all the previous year balances to the current year balances. In some package, there is no need to add another balance transfer entry, they automatically get carried forward in the current year. These balances will stay in the books indefinitely until some accounting entry is made.

What is the purpose of posting of general journal & other journal?

General journal posting

This transaction should be posted to the cash receipts journal - not general journal.