In a small bowl or measuring cup, whisk together the soy sauce, lime zest and juice, ginger, and brown sugar. Place the salmon fillets in a shallow pan or baking dish and pour the marinade over the top. Cover with plastic wrap and marinate in the refrigerator for at least 30 minutes and up to 2 hours.
Transfer the salmon to the Instant Pot®, skin side up, and pour the remaining marinade over the fish. Lock the lid in place and turn the valve to Sealing. Press the Pressure Cook button and set the cook time for 1 minute (for rare salmon) or 2 minutes (for medium salmon) at low pressure.
Let the steam release naturally for 5 minutes, then turn the valve to Venting to quick-release any residual steam. Carefully remove the lid and use tongs to transfer the salmon to a serving platter. Press the Cancel button to reset the program.
Press the Sauté button and simmer the cooking liquid until it has thickened, 3–4 minutes. To serve, spoon over the salmon and top with sesame seeds. Serve rice alongside.