They are just going to continue to rust/corrode and stain the metal and the trim beneath until either a) the nails rust away to nothing or b) you replace the nails with the correct nails. Your choice of when...................................

Rust converter. That's converter, not aresstor, not paint, not remover,,,,,,C_O_N_V_E_R_T_O_R.

My all time favorite is Jasco, 1/2 pint will do all those nails with some to spare so only buy a quart. Follow directions on container by using a medium size artist brush or oversize Q-tip type applicator to "DAB" product on each nail. Paint over it within 48 hours. Use same or similar applicator to DAB aluminum paint on to nails, then a secound or third coat. If desired, you may paint flashing with aluminum paint or if you want flashing to be colored, prime flashing and converted nails before applying colored paint.

Two others in case you don't find Jasco, Coroseal and Gemplers. There's others, just be sure it's a major brand name.

Replace them with galvanized or stainless steel.

The root cause is likely water sitting on the flashing. Check if the flashing has positive drainage away from the house. If it's pitched inward or flat, water pools and rusts the nails. Fixing the drainage might be more important than treating the rust. Otherwise you'll be doing this again in a few years.

The problem is there should NOT be any nails on the face of a drip cap. No matter the rust, each one is a leak point. This was DIYed, you can tell they did not have a brake to form the drip cap.