Calculating the partial sum of a expl3 sequence

It's more natural to use expl3 iteration over the seq I think:

\documentclass{article}
\usepackage{xparse}

\ExplSyntaxOn
\seq_new:N \g_tmp_seq
\seq_new:N \g_partialsum_seq
\seq_set_from_clist:Nn \g_tmp_seq {3, 1, 2, 4, 0, 1}
\seq_set_from_clist:Nn \g_partialsum_seq {0}
\tl_new:N\l_sum

\seq_map_inline:Nn\g_tmp_seq{
  \tl_set:Nx\l_sum{\int_eval:n{\l_sum+#1}}
  \seq_gput_right:NV \g_partialsum_seq \l_sum
}

\NewDocumentCommand\partialsum {m}{
  \seq_item:Nn \g_partialsum_seq { #1 }
}
\ExplSyntaxOff

\begin{document}
\partialsum{1} % should be 0

\partialsum{2} % should be 3

\partialsum{3} % should be 4

\partialsum{4} % should be 6

\partialsum{5} % should be 10

\partialsum{6} % should be 10
\end{document}

I dropped calc and forloop entirely, as you can do everything using plain expl3. As a bonus you can make the command expandable and have an optional argument to set the starting point of the summation.

The major difference from your version is that the code doesn't store the summed sequence. Each time the command is used the summation is done for the selected interval.

\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn
\seq_new:N \g_noibe_numbers_seq
\seq_set_from_clist:Nn \g_noibe_numbers_seq {3, 1, 2, 4, 0, 1}
\NewExpandableDocumentCommand \partialsum { O{1} m }
  { \__noibe_partialsum:Nnnn \g_noibe_numbers_seq {#1-2} {#2-2} { } }
\cs_new:Npn \__noibe_partialsum:Nnnn #1 #2 #3 #4
  {
    \int_compare:nNnTF {#2} > {#3}
      { \int_eval:n {#4 0} }
      {
        \exp_args:Nf
        \__noibe_partialsum:nN { \int_eval:n {#2+1} } #1
          {#4} {#3}
      }
  }
\cs_new:Npn \__noibe_partialsum:nN #1 #2
  {
    \exp_args:Nf
    \__noibe_partialsum:nnNnn { \seq_item:Nn #2 {#1} + } {#1} #2
  }
\cs_new:Npn \__noibe_partialsum:nnNnn #1 #2 #3 #4 #5
  { \__noibe_partialsum:Nnnn #3 {#2} {#5} {#4#1} }
\ExplSyntaxOff
\begin{document}
\partialsum{1} % should be 0

\partialsum{2} % should be 3

\partialsum{3} % should be 4

\partialsum{4} % should be 6

\partialsum{5} % should be 10

\partialsum{6} % should be 10

\ifnum\partialsum{5}=\partialsum{6}
  The fifth item is zero.
\else
  The fifth item is nonzero.
\fi
\end{document}

Much less code with expl3:

\documentclass{article}
\usepackage{xparse}

\ExplSyntaxOn
\NewDocumentCommand{\setsequence}{m}
 {% #1 = items
  \seq_set_from_clist:Nn \l_noibe_sequence_seq { #1 }
 }

\NewExpandableDocumentCommand{\partialsum}{m}
 {% #1 = (one more than the) number of items to sum
  \int_eval:n
   { 
    0 \int_step_function:nN { #1 - 1 } \__noibe_sequence_sum:n
   }
 }

\cs_new:Nn \__noibe_sequence_sum:n
 {
  \int_compare:nF { #1 > \seq_count:N \l_noibe_sequence_seq }
   {
    + \seq_item:Nn \l_noibe_sequence_seq { #1 }
   }
 }
\ExplSyntaxOff

\begin{document}

\setsequence{3, 1, 2, 4, 0, 1}

\partialsum{1} --- should be 0

\partialsum{2} --- should be 3

\partialsum{3} --- should be 4

\partialsum{4} --- should be 6

\partialsum{5} --- should be 10

\partialsum{6} --- should be 10

\partialsum{7} --- should be 11

\partialsum{8} --- should be 11

\end{document}