Try this test:
azqhazqhazqhazqhazqhazqhazqhazqhazqhazqhazqhazqh

My answer:
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++.+++++++++++++++++++++++++.<+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++.<++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++.<+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++.>>>.<.<.<.>>>.<.<.<.>>>.<.<.<.>>>.<.<.<.>>>.<.<.<.>>>.<.<.<.>>>.<.<.<.>>>.<.<.<.>>>.<.<.<.>>>.<.<.<.>>>.<.<.

I have WA 28 Also.
First 27 test I have passed with N=3 using Dejkstra.
But N==4 means very many possible states 27*27*27*27*4*50
and unprocessing.

my WA on test 30 was because, I forgot:
"If the pointer before increment points to the rightmost byte of the array, then after increment it points to the leftmost byte."
and
"If the pointer before decrement points to the leftmost byte of the array, then after increment it points to the rightmost byte."

In the statement it is said, that you may assume to use at most 4 bytes, but it is not said WHICH four bytes.. In the case of test 30, the best four are the rightmost one and the three leftmost.

However, now I'm stuck on 31...

no. Output of "aba" don't use two cells. You should use one.
after you output 'a' do next:
+.-.
to output 'b' and return back to 'a'. Your task is to minimalize output length

The statement is ambiguous. The input string contains only letters ('a'-'z') and the array of 30000 memory cells is full of 0. It seems that all the cells should be modified. Please clarify this.