If $|U|=8$ then there are $2^8=256$ different subsets of $U$, say
$X_1,\dotsc,X_{256}$. Let $s_i$ be the sum of the numbers in
$X_i$. This is the sum of at most $8$ numbers from $1$ to $32$,
so the largest possible value is $25+26+\dotsb+32=228$. We
therefore have $256$ numbers $s_i$ in the range
$\{0,\dotsc,228\}$, so they cannot all be different. We can
choose $i<j$ with $s_i=s_j$ and then the sets $A=X_i$ and $B=X_j$
are distinct but have the same sum. The sets
$A'=A\setminus(A\cap B)$ and $B'=B\setminus(A\cap B)$ are then
distinct disjoint subsets of $U$ with the same sum.