JavaScript (Node.js),  608 597 586  551 bytes

Takes input as (source)(input), where input is an array containing single characters and/or integers.

s=>I=>eval([..."@CDEJKLMNQTUWYZ^_bkmqw"].reduce((s,c)=>(a=s.split(c)).join(a.pop()),"(R=X=>(H={},o=P=[],S=[],z=x=p=i=0,gUs[p]?~(j=` 	\n`.indexOf(s[p++]N?j:gK:3,hUgK<2?Mn*2+j):V=n,GUx=z--&&S.pop(y=x),FUg(x=`mbQQ$mS[z+~(b])TSCce((z-=V=b-1,V)$mkJLmyJkTTLH[x]=y$mHD)qR[M1)]=_?R[P@p),V]:_Y_&!kY_&k<0Yp$E?P.popK:pT$^QQQQQqW+yw-yw*yw/y|0w%yN:^QTTo+=Buffer(D)$o+=k$Z.codePointAtKZ`Ct`$`[n-9],x&&eval(x,n=1N<3&&F(n*3+jN(1N(0)||R(1)||owN:^$WqTT$mz=S@kGKbgK?-M0):M0N_p$M1),E^E?g:pZ$HD=I[i++]Y?R[V]:Wz>1?(LmxU=n=>T$$QqqN))Mh(Lk,k,K()J),mx)$Ep=XD[k][email protected]"))

Try it online! (all test cases without comments)

Try it online! (an example with a commented source code as input)

How?

Because the code is packed with RegPack, it is more efficient to repeat the same pieces of code over and over rather than defining too many helper functions.

The sequences of spaces, tabs and linefeeds are converted to an integer in base 3 with a leading \$1\$ until they match a known value:

 seq. | mnemonic | base 3 | decimal
------+----------+--------+---------
 SS   |    PSH   |    100 |     9
 STS  |    CPY   |   1010 |    30
 STN  |    SLI   |   1012 |    32
 SNS  |    DUP   |   1020 |    33
 SNT  |    SWP   |   1021 |    34
 SNN  |    DIS   |   1022 |    35
------+----------+--------+---------
 TTS  |    PUT   |   1110 |    39
 TTT  |    GET   |   1111 |    40
------+----------+--------+---------
 NSS  |    LBL   |   1200 |    45
 NST  |    JSR   |   1201 |    46
 NSN  |    JMP   |   1202 |    47
 NTS  |    JZE   |   1210 |    48
 NTT  |    JMI   |   1211 |    49
 NTN  |    RTN   |   1212 |    50
 NNN  |    END   |   1222 |    53
------+----------+--------+---------
 TSSS |    ADD   |  11000 |   108
 TSST |    SUB   |  11001 |   109
 TSSN |    MUL   |  11002 |   110
 TSTS |    DIV   |  11010 |   111
 TSTT |    MOD   |  11011 |   112
------+----------+--------+---------
 TNSS |    CHR   |  11200 |   126
 TNST |    INT   |  11201 |   127
 TNTS |    RCH   |  11210 |   129
 TNTT |    RNU   |  11211 |   130

The commands are simply stored in a lookup table and executed with eval().

Everything is wrapped within the function \$R\$ which is called twice:

• First pass (\$X=0\$): the jumps and the stack errors are ignored so that the whole code is parsed linearly and all labels are stored
• Second pass (\$X=1\$): the code is executed normally

Unpacked and formatted

s => I => (
  R = X => (
    H = {},
    o = P = [],
    S = [],
    z = x = p = i = 0,
    g = n => s[p] ? ~(j = ` \t\n`.indexOf(s[p++])) ? j : g() : 3,
    h = n => g() < 2 ? h(n * 2 + j) : V = n,
    G = n => x = z-- && S.pop(y = x),
    F = n => g(
      x = [
        /* PSH */ "z=S.push(g()?-h(0):h(0))",,,,,,,,,,,,,,,,,,,,,
        /* CPY */ "z=S.push(S[z+~(g()?-h(0):h(0))])",,
        /* SLI */ "S.splice((z-=V=g()?-h(0):h(0))-1,V)",
        /* DUP */ "z=S.push(G()),z=S.push(x)",
        /* SWP */ "G(),G(),z=S.push(y),z=S.push(x)",
        /* DIS */ "G()",,,,

        /* PUT */ "G(),G(),H[x]=y",
        /* GET */ "z=S.push(H[G()])",,,,,

        /* LBL */ "R[h(1)]=p",
        /* JSR */ "h(1),p=X?R[P.push(p),V]:p",
        /* JMP */ "h(1),p=X?R[V]:p",
        /* JZE */ "h(1),p=X&!G()?R[V]:p",
        /* JMI */ "h(1),p=X&G()<0?R[V]:p",
        /* RTN */ "p=X?P.pop():p",,,
        /* END */ "p=X?g:p",,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

        /* ADD */ "z>1?(G(),G(),z=S.push(x+y)):p=X?g:p",
        /* SUB */ "z>1?(G(),G(),z=S.push(x-y)):p=X?g:p",
        /* MUL */ "z>1?(G(),G(),z=S.push(x*y)):p=X?g:p",
        /* DIV */ "z>1?(G(),G(),z=S.push(x/y|0)):p=X?g:p",
        /* MOD */ "z>1?(G(),G(),z=S.push(x%y)):p=X?g:p",,,,,,,,,,,,,,

        /* CHR */ "o+=Buffer([G()])",
        /* INT */ "o+=G()",,
        /* RCH */ "H[G()]=I[i++].codePointAt()",
        /* RNU */ "H[G()]=I[i++]"
      ][n - 9],
      x && eval(x, n = 1)
    ) < 3 && F(n * 3 + j)
  )(1)
)(0) || R(1) || o