Is it possible to downgrade a business/first class flight to economy without the ticket purchaser knowing?

Downgrading a flight maybe possible. It would depend on whether your ticketed fare allows you to do that. However, you may have issues if it was bought by a company as they may have used their corporate accounts and refunds (and partials) would go there. Even if it didn't they would notice they received less corporate air miles (or equivalent depending on the airline) for your flight compared to what they expected. And finally if they requested the ticket copy, it will show the fare class you flew especially on your phone as it would be on your phone. You maybe able to bypass that last one by printing out the boarding pass first (as higher classes generally allow earlier check-in) then changing the ticket.

You maybe able pull a sly one on your uncle unless he checks your booking online. Which the company could do too.

It wouldn't be the best idea to downgrade a flight without informing the purchaser first.

But you feel like this is either a waste of money

It's a) not your money, b) already spent , c) the airlines need the cash.

or you're petite and don't like the larger seats

Walk into economy, find someone you like the looks of, trade seats.

If you want to be cruel, find a mother with an infant.

or you want to minimize your carbon footprint,

Won't change. The carbon footprint stays the same whether the plane is empty, half full, or stuffed to the rafters with tribbles. Where you actually sit is irrelevant.

Most airlines with online booking ( I think that's everyone except the dodgier African carriers) let anyone with the reservation code and the passenger's name look at (but not change) the reservation details. They would see. And unless it's a cash ticket (highly unlikely) any refund would go back on the purchaser's card, not yours.

So, it seems to really be a question of "what do you do when they find out".

It really depends on how the "ticket" is provided to you but assuming it is an airline printed ticket rather than an e-Ticket;

A lot of tickets have AGT/REF endorsements which means any refund must go back to the purchaser via the agent.

However, instead of a refund, you could ask the airline to re-issue your lower class ticket and issue the difference in Miscellaneous Charge Orders (MCOs) which are the airlines equivalent of credit notes/gift vouchers. MCOs are generally only issued by IATA airline members. MCOs can be used for part-payment of flights, excess baggage etc

Unless the original ticket was issued with an airline specific restriction, MCOs are also valid to redeem with any other IATA airline!!

Lastly, flexible airline tickets are usually valid for 12 months but you can extend their validity by having the ticket re-issued before their expiry.

Summary

Your scheme is absolutely, definitely, 100% impossible, because: any excess amount refunded must and can only go to the purchasor, the rich relative.

So that's that.

Note however that regarding the first-explained aspect of your Caper, the answer is positively yes. A primary reason (rich!) people give 1st class tickets as presents is that they are totally flexible. If one is ever lucky enough to get a 1st class ticket somewhere as a present (it's certainly never happened to me, tragically!) you can just call the airline and change anything you want. Yes sir, no sir, three bags full sir.

However, here's some astounding information on this:

Thanks to DC below, it appears that some (all?) airlines now offer "cheap, incredibly crappy" first class fares. in the Lufthansa example, I checked near dates and they only offered what (I would have thought of as) ordinary 1st class fares (to wit: obviously one can utterly freely change anything one wishes, at any time). However, a couple of weeks out, you can buy these "Easyjet-like" first class fares! :) which have restrictions and change fees and the like. So, consider this factor when implementing the Caper described by the OP.

Note that, with for example a €6000 "real" 1st class ticket given to you by the rich relative, after you refunded it, paid the €500 fee, and bought a €308 cheap ticket on a discount airline instead, you'd still be ahead €5000 in this "caper" suggested by the OP.

But again — very simply — all refunds today go only to the purchasing credit card, so the Caper wouldn't work. Thanks again, DC, for this "first class information"!

Even if you can't get the money, you can almost surely opt to sit in economy seating. Just get to the boarding area early and inform them you'd rather be seated in economy, and I'm sure they'll be happy to open up another seat they can charge somebody else a fee to "upgrade" to.