There is the Presto card system, which you can get at the GO station, but not all subway stations currently accept it (many a tourist would use do). All streetcars apparently now do accept it. If you are primarily visiting the downtown core, you'll probably be taking the subway and streetcars, and less so buses. At subway stations you can usually pay in cash (and get change) if you need to, though the fare is higher. Streetcars can accept exact fare change.
It's a a stored-value card- you can charge it with a credit card at the ticket window of a GO station. It is zone system so you must tap it on entry and exit or you could be charged more than you trip should cost (if you miss the exit tap) or fined (if you are riding effectively without a ticket).
See the FAQ for information on refunds- I only have direct experience charging it for GO bus trips for my son. It seems to say that you could get an immediate refund of less than $50 balance, less 4% service charge, otherwise you might have to wait for months.
The TTC also has a day pass for $12, which you could consider. The TTC fare using Presto is 2.90 CAD vs. 3.25 cash. Presto also gives you a ~11% discount on the GO train fares (probably around $10 each way from Burlington to Union Station).
You can also buy TTC tokens at 3 tokens for $8.70 (2.90 per trip if you use them all). They're really light and small- like an aluminum dime, take care they don't get lost.
Personally, for a day or two, I would suggest buying a return ticket for cash or credit card at the GO station and picking up some tokens (credit and debit accepted over $10) at the subway station- and keeping it simple. (Or use Presto and load it with the GO fare plus $12 or so for the day).
For future reference, TTC tokens are due to be eliminated in favor of Presto, but that will not happen until mid-2017.
Edit: With regard to borrowing your host's Presto card, yes it is explicitly allowed and would be an excellent solution:
Can more than one person use the same PRESTO card to travel at different times?
Yes, PRESTO cards are transferable. If you are using a card that has a concession fare set, you must also be eligible for any discounted fares set on the card (i.e. child, student, senior).
A student, senior or child may use an adult's PRESTO card to travel at full adult fare. However, an adult may not use a student’s, child’s or senior’s PRESTO card to travel.
Also, there is no problem at all using your credit card to refill your host's Presto.