User has set the income and expense accounts for expenses. When are the expense entries actually posted to these accounts and what determines whether they are posted to the income account or expense account?
When you link an expense entry to a vendor bill, it gets posted to the expense account specified for it (Default Item Expense Account, if none is specified). In case of employee reimbursable expenses, they get posted to the expense account when you approve them. In both cases, the expense entry becomes a liability that you need to pay off. So it is posted to the expense account and it credits (increases) your Accounts Payable. The billable status of the expense entry does not change this. It only affects the A/R side.
Similarly, when you create an invoice from a billable expense entry, it gets posted to the income account specified for it (Default Item Income Account if none is specified). It is a potential source of income, so it is posted to the income account and it debits (increases) your Accounts Receivable.
For approved non-billable entries, no accounting is generated on the A/R side, although they reflect on the account snapshot of the income account as a zero value entry. They do not contribute to the potential income, so they do not debit A/R. But if they are reimbursable, they get posted to the expense account and credit A/P, as described above.