Shamir's Secret Sharing

Jelly, 15 bytes

251©xX€⁵0¦ḅЀ%®

Expects t, n, and s as command-line arguments. Try it online!

How it works

251©xX€⁵0¦ḅЀ%®  Main link. Left argument: t. Right argument: n Third argument: s

251©             Yield 251 and copy it to the register.
    x            Repeat [251] t times.
     X€          Random choice each; pseudo-randomly choose t integers from
                 [1, ..., 251]. Since 251 = 0 (mod 251), this is equivalent to
                 choosing them from [0, ..., 250].
       ⁵0¦       Replace the last generated integer (index 0) with s (⁵).
          ḅЀ    Interpret the resulting array as a base-k number, for each k in
                 [1, ..., n], and convert to integer.
              ®  Yield 251 from the register.
             %   Take the generated integers modulo 251.

Mathematica, 59 56 bytes

Mod[Power~Array~{#2,#-1}.RandomInteger[250,#-1]+#3,251]&

Takes three arguments in the order t, n, and s. Constructs a 2d-array with n rows and t-1 columns. Each row vector j, numbered from 1 thru n, contains the powers of j thru j^t-1. Then a vector of random integer coefficients in the range 0 thru 250 is created with t-1 values. That is matrix-multiplied with the 2d-array, and then s is added element-wise and taken module 251 to get the value of the polynomial at each of the n points.

CJam, 27 25 bytes

{,:)@({;251mr}%@+fb251f%}

Test it here.