Algebra

[static gyro]

Try to determine a few function values with m=n=0; n=0; m=1...

$$m=n=0 : f(f(0)+0)+f(0)=f(0)+f(3\cdot 0)+2014 \Rightarrow f(f(0))=f(0)+2014$$
$$m=n ; f(f(m)+m)+f(m)=f(m)+f(3m)+2014 \Rightarrow f(f(m)+m)=f(3m)+2014$$
$$n=f(m) : f(f(m)+n)+f(m)=f(2 f(m)) + f(m) =f(f(m))+f(3m)+2014$$ $$
= 2\cdot (f(3m)+2014)$$

[errant spruce]

Pfötchen

[static gyro]

$$n=0,m\in Z : f(f(m)+0) + f(m) = f(f(m))+f(m)=f(0)+f(3m)+2014$$
$$\Rightarrow [ f(3m)+2014 ] + f(m) = f(0) + [ f(3m)+2014 ] $$
$$\Rightarrow \forall m\in Z : f(m) = f(0) $$

[errant spruce]

Pfötchen

[static gyro]

Sorry second equation is wrong !!!