No, it does not mean "I'll be sad until I die." The word ἕως ("until") is used here as a preposition rather than a conjunction, as pointed out in BDAG sub-entry #5:
marker of degree and measure, denoting the upper limit, to the point of ... Mk. 14:34 ...
The phrase then means "sorrowful to the point of death". This remains less than perfectly perspicuous, but the oddity of expression is helpful in that it suggests an allusion to OT texts whose language Jesus here echoes. The closest verbal parallel to the phrase of interest is Jonah 4:9:
Σφόδρα λελύπημαι ἐγὼ ἕως θανάτου.
I am exceedingly grieved, unto death.
The Greek closely follows the Hebrew, itself not a common expression, appearing nowhere else in the Hebrew Bible. Commentators are divided about whether it should be taken as hyperbolic ("sad enough to die") or an unusual superlative ("exceedingly sad"), but the sense is clear enough.
In context, Mark 14:34 is further clarified by its allusion to LXX Psalms,1 particularly Psalm 41:6a, 12a (= EVV 42:5, 11; Heb. 42:6,12), and 42:5a (= EVV and Heb. 43:5), which are identical:
מַה־תִּשְׁתּ֬וֹחֲחִ֨י ׀ נַפְשִׁי וַתֶּהֱמִ֪י עָ֫לָי
Why are you cast down, O my soul, and why are you in turmoil within me?
ἵνα τί περίλυπος εἶ, ψυχή, καὶ ἵνα τί συνταράσσεις με;
The idea is that of overwhelming grief.2
1. The Greek περίλυπος ("sorrowful") is an uncommon word, used only 7 times in the LXX; only the three (identical) verses mentioned here have a referent ψυχή ("soul").
2. For a survey of the interpretive options and a different conclusion, see W. L. Holleran (The synoptic Gethsemane. A critical study) who sees here a request by Jesus to die, here and now, and thereby escape the torment of the cross. A related take (cited there) that doesn't go quite so far is that of Bultmann (TDNT, IV [Grand Rapids, 1967] 323), who sees the "obvious" meaning: "to be so full of sorrow that I would rather be dead". Nevertheless, the English language commentaries mostly fail to see a death wish here (Cranfield, Evans, France, Marcus).