[HerpDerp]

Can this be done with the Birch reduction, catalytic hydrogenation or the Clemmensen reduction?

[ooosh]

No, you can not do it in one step,the methods you listed can not give you an alkane either . 
You may reduce the carboxylic acid to an alcohol(use LiAlH4),then halohydrocarbon,then dehalogenation to alkane.

[SVXX]

What about decarboxylation reactions? RCOOH can be converted to R by reacting it with NaOH and CaO. For example,

HCOOH ----> H₂ + CO₂
CH₃COOH -----> CH₄ + CO₂

This method is elaborated further in http://www.chemguide.co.uk/organicprops/acids/decarbox.html . For other types of decarboxylations, check out the Wikipedia article.
Another important fact that you must know is that certain acids, such as beta-keto acids, decarboxylate simply on warming the compound. This is because of the formation of a favourable six member transition state on warming.