Was birgst du so bang dein Gesicht
is a token of a German idiom which can be found pretty often in literatur from the older days. But the idiom can still be heared today even in everyday language.
The was here is not used in the otherwise usual sense of asking for an object of a verb as in
Was isst du da? - Eine Banane, das siehst du doch!
In your sentence, the was has a different meaning. See the following examples:
Was bohrst du dir dauernd in der Nase?
Was nestelst du immer an deinem Regenschirm herum?
Was bummelst du schon wieder so rum?
Was sendet der Trump da immer diese Twitter-Nachrichten, hat er nichts besseres zu tun?
Was regnet es heute schon wieder ohne Unterlass?
In all these cases, the "was" is used as a shorter synonym for
Warum um Himmels willen...
which is like warum plus disapprobation of the behaviour or phenomenon in focus.
An interesting case is
Was isst du denn?
Here, the was could be read as asking for an object; or it could be interpreted as
Warum isst du denn schon wieder?
Was isst du denn schon wieder eine Banane? Du hast doch heute schon drei gegessen!
Things depend here simply on intonation (and other means of non-verbal communication accompanying the verbal utterance).
Is it provocative?
In certain situation asking things like
Was kuckst du mich so an?
can be perceived as provocative (as RoyPJ in his answer correctly noted). However, this depends more on the situation as such (in the above example the question would be provocative in most cases because asking somebody why he is looking at you is in most cases provocative independent of wording). But there are also cases where a was-disapprobation would not be provocative but simply carry the message of emotional involvement:
Was regnet es heute nur den ganzen Tag!
Was würde ich gerne nochmals nach Venedig fahren!
Oh. The last one even shows that was may also replace wie, since warum nur is not what the person speaking intends to say; and that it also may be used to express a wish, not only dissaproval.