EDIT: The answer below seems to be incorrect, as I seem to have read the results of the paper (mentioned below) too superficially. :EDIT
If I understood correctly, the following paper shows (among other things) that FVS is solvable in polynomial time on graphs of maximum degree at most 3:
Cao, Yixin; Chen, Jianer; Liu, Yang (2010), On Feedback Vertex Set: New
Measure and New Structures in Kaplan, Haim, "SWAT 2010", LNCS 6139: 93–104.
They study a slightly more general problem, namely DISJOINT-FVS, where in
addition to the parameters of FVS, two certain vertex sets V_1 and V_2 are
given. For details, see the paper. Then, ordinary FVS instances are a special
case of DISJOINT FVS instances where the given set V_2 equals the entire vertex
set, and V_2 is empty. They show roughly the following:
a) DISJOINT FVS can be reduced to DISJOINT FVS on graphs of minimum degree at
least 3 without increasing the maximum degree.
b) On 3-regular graphs, DISJOINT FVS instance can be solved in polynomial time.
Together, these two results show that FVS on graphs of maximum outdegree 3 can
be solved in polynomial time. Beware that I have not read the paper very
thoroughly, and may have misunderstood or misstated something, or even everything.
a) This is oversimplified. Only the maximum degree in the graph induced by V_1
is not increased.
b) Also oversimplified. If the graph induced by V_1 is 3-regular, then this
DISJOINT FVS instance can be solved in polynomial time.