In any interplanetary mission, it's impossible to perfectly accurately measure the position and velocity of a spacecraft or a planet, so mission planners generally schedule a number of opportunities to measure the accumulated error and make a course correction. Therefore, there's no reason a Mercury probe mission would have been a failure if General Relativity wasn't understood.
The amount of precession of Mercury's orbit due to GR is, according to Wikipedia, 43 arcseconds per century. Let's see how that would impact a mission like MESSENGER.
MESSENGER took about four years to make its first encounter with Mercury. If, for some reason, you didn't notice that Mercury's actual precession didn't match up with your theory, and planned to go where you thought Mercury should be according to theory, Mercury's orbit would be rotated 1.72 arcseconds out of position. At Mercury's distance from the Sun, that amounts to an error of about 500km.
However, MESSENGER did a flyby of Earth and two of Venus to reach its Mercury-bound trajectory. After the second Venus flyby, it made a fairly large course correction (called DSM-2) to properly line up the Mercury encounter, and this occurred only three months before reaching Mercury. So if you measured Mercury's position at that point, you'd be just 0.1 arcseconds off target when aiming three months out, missing your target by only 30km.
MESSENGER's first flyby of Mercury had a closest approach of 200km, so 30km wouldn't risk a crash or a complete miss; uncorrected, though, it would mess up the timeline of the orbital capture and insertion. As it happens, MESSENGER did a small correction a month after DSM-2. Assuming, again, our Relativity-ignorant mission controllers just corrected for Mercury's observed position at that time, the error would come down to about 20km.
Between that point and the first Mercury encounter, MESSENGER used a different strategy for the very small course corrections needed, in order to save fuel: it angled its solar panels slightly to use them as sails on the solar wind! The remaining error due to General Relativity at this point could be corrected with a delta-V of only about 4mm/s; if solar panel sailing wasn't sufficient, the small attitude thrusters on the spacecraft would be. So I don't see any reason MESSENGER would have failed given an ignorance of GR; at worst it would have spent a negligible amount of additional propellant in corrections, possibly shortening its 4-year lifespan in Mercury orbit by a couple of months.
The only other mission to reach Mercury was Mariner 10; it also did a single Venus flyby for gravitional assist. It did have maneuvering thrusters, and it was nearly two months between its Venus flyby and first Mercury flyby (at 700km), so similar assumptions would hold.
Finally: in 1859, Le Verrier measured the precession effects of GR, without knowing what the cause was; he got a figure of 38 arc-seconds per century instead of 43. If our mission plan took that figure into account, the GR error figures I've given here would be much reduced: ~60km at 4 years, ~3.5km at 3 months, ~2.5km at 2 months.