Arriving Heathrow airport terminal 2, departing terminal 3, but need lounge access in terminal 2

I don't believe so, no, unless you have arrivals lounge access.

When you get off your flight in your original terminal (in this case Terminal 2), you'll be in the arrivals passenger flow. Unlike a domestic US arrival, for example, you don't just get dumped out into the terminal's departure gate area, but into a secure area. You'll then follow the purple Flight Connections signs, which for most of the way will be the same as the yellow Arrivals and Baggage Claim flow.

Once you get to flight connections, if you had an onward flight from the same terminal (Terminal 2), you'd go through security there, then into the departure gate area. However, you won't have a boarding pass for that terminal, so they won't let you into the flight connections security for that terminal. Instead, you'll take the Flight Connections transfer bus over to your next terminal (in this case Terminal 3; there's one bus per terminal, well signposted), then on arrival there, clear security and enter the terminal.

If you have arrivals lounge access, and suitable permission to enter the UK, you could follow the yellow Arrivals process, clear UK immigration, exit through customs, and go to the landside lounge. After that, you should follow the signs to the Heathrow Express and Underground down into the tunnels, and then signs to Terminal 3. It'll take you about 10-15 minutes to get from Terminal 2 to 3, depending on walking speed and familiarity with the terminals.

Terminal 3 does have a paid-for lounge, which is quite nice, so you could just rock up there and pay for entry. It's the No.1 Traveller.

Heathrow used to offer a poorly-advertised ability for any connecting passenger with a very long layover to go to Terminal 3 to access the paid-for lounge there. It meant re-clearing security twice, and following a not-great-signed route to leave for Terminal 3, so it wasn't well known. The last time I checked (about 2 months ago), they hadn't changed it to also offer the same for the paid-for lounge in Terminal 2, so I believe Terminal 2 entry still requires a Terminal 2 departing boarding pass for that day. Even if you could get back into Terminal 2 airside after security, I don't think it'd help as you wouldn't have a Star Alliance same-day departure from that terminal, so I don't believe you'd be allowed into the lounge anyway.

Star Alliance has lounges on the arrival level at Heathrow Terminal 2. I don't know the details of your lounge access, but I don't see any reason why you shouldn't be able to use them. Of course you will still need to transfer to T3 at a reasonable time.