What is the easiest way to concentrate a solution without causing it to hydrolyze? I’ve been working with iron acetate solutions in dilute acetic acid (vinegar) and I’ve been running into a problem. Every time I try to boil off the solvent the solution hydrolyzes to form iron oxides. I’ve given up trying to dry it completely because supposedly iron acetates can’t exist outside of solution, but is there a way of concentrating the solution?
Try adding Na2SO4 to the mix and freeze forming Na2SO4.7H2O. Slow, but at least, no oxygen-related chemistry, discussed below.

As a sidebar, you likely have formed basic ferric acetate from boiling, which adds oxygen, to your acidic (vinegar) and aqueous ferrous acetate mix. 

Likely reaction mechanics are complex, see, for example, Table S1 here . Per the source my take of a possible reaction sequence:

$\ce{ O2 (aq) + Fe^{2+} → Fe^{3+} + .O2− }$ ( moderately fast) )

$\ce{ . O2− + Fe^{2+} + 2 H+ → Fe^{3+} + H2O2}$  (fast )

$\ce{ H2O2 + Fe^{2+} → Fe^{3+}  + .OH + OH− }$ ( slow)

$\ce{  .OH + Fe^{2+} → Fe^{3+} + + OH− }$ ( fast )

Net radical reaction: 

$\ce{ O2 + 4 Fe^{2+} + 2 H+ → 4 Fe^{3+} + 2 OH-}$ 

The creation of FeOOH is also possible in less acidic conditions per his source: 'Air Oxidation of Ferrous Iron in Water'.