Does money actually change hands trading futures?

Assuming that the asset underlying a futures contract pays no dividends or associated (storage, etc) costs, I have the following formula for the price F_t of a futures contract at time t:

F_t = S_t * e^{r (T-t)}

where S_t is the value of the underlying asset at time t, r is the risk-free rate, and T is the contracts delivery date.
Suppose that F_t < 0. If I were to take a long position on this contract at time t in a real world situation, would I immediately receive the amount F_t, or would all money change hands only at delivery time T?

I don't understand your question but just as a note F_t cannot be negative because a price (S_t) is always positive and e^x > 0 for any real x. The product of two positive numbers is positive.

Sorry, my mistake. I meant to say that if you too a short position on the contract (i.e. agreed to make delivery of the underlying asset), in which case the value of the contract (to you) would always be negative?

Additionally, unlike an options contract, surely the value of a future may be positive or negative - since the payoff may be either. If, upon the delivery date, the price of the underlying asset has fallen, you would be obliged to pay the difference? Does this mean that the formula stated above is incorrect? Should it instead have been

V_t = F_0 - S_t * e^{r (T-t)}

Price is always positive. Value might be negative. Everything is correct: V_t = F_0 - F_t = F_0 - S_t * e^{r (T-t)} (for long position). It will be -V_t for short in this case.
That is how most of the futures work these days, you pay the difference which is more convenient.