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I don't know what solve is doing (with inequalities/assumptions), but here your system of polynomial equations has finitely many solutions, so you can filter out those satisfying the inequalities:

R.<x,y,M,a,b,c,d,A,B,C,D,t,u,v,S,T,U,V> = PolynomialRing(QQ)

eq1 = 3*(((2*3367-3*y+1)/24)+3*x*(x+1)/2)+1-V  
eq2 = -3367+3*x*(x+1)/2-3*y*(y-1)/2+(3*x+1)*(3*x+2)/2  
eq3 = 3*(((2*V-3*v+1)/24)+3*x*(x+1)/2)+1-U  
eq4 = V+3*v*(v-1)/2-12*x*(x+1)/2-1  
eq5 = 3*(((2*U-3*u+1)/24)+3*x*(x+1)/2)+1-T  
eq6 = U+3*u*(u-1)/2-12*x*(x+1)/2-1  
eq7 = 3*(((2*T-3*t+1)/24)+3*x*(x+1)/2)+1-S  
eq8 = T+3*t*(t-1)/2-12*x*(x+1)/2-1  
eq9 = 3*(((2*S-3*1+1)/24)+3*x*(x+1)/2)+1-S  
eq18 = -M+9*((2*3367 - 3*y + 1)/24+(y-1)*(y+1)/8)+1
eq19 = 3*((2*M - 3*(x+1) + 1)/24 + (3*x*(x + 1))/2) + 1 - A
eq20 = -M + (3*x*(x + 1))/2 - (3*x*(x + 1))/2 + ((3*x + 1)*(3*x + 2))/2
eq21 = 3*((2*A - 3*a + 1)/24 + (3*x*(x + 1))/2) + 1 - B
eq22 = A + (3*a*(a - 1))/2 - (12*x*(x + 1))/2 - 1
eq23 = 3*((2*B - 3*b + 1)/24 + (3*x*(x + 1))/2) + 1 - C
eq24 = B + (3*b*(b - 1))/2 - (12*x*(x + 1))/2 - 1
eq25 = 3*((2*C - 3*c + 1)/24 + (3*x*(x + 1))/2) + 1 - D
eq26 = C + (3*c*(c - 1))/2 - (12*x*(x + 1))/2 - 1
eq26 = 3*((2*D - 3*d + 1)/24 + (3*x*(x + 1))/2) + 1 - S
eq27 = D + (3*d*(d - 1))/2 - (12*x*(x + 1))/2 - 1
eq28 = 3*((2*S - 3*1 + 1)/24 + (3*x*(x + 1))/2) + 1 - S
eq29 = x-24

I = R.ideal([eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8,eq9,eq18,eq19,eq20,eq21,eq22,eq23,eq24,eq25,eq26,eq27,eq28,eq29])
nonneg = [x-1, y-1]
pos = [A-M, B-A, C-B, D-C, S-D, V-3367, U-V, T-U, S-T]
for sol in I.variety():
    if all(lhs.subs(sol) >= 0 for lhs in nonneg) and all(lhs.subs(sol) > 0 for lhs in pos):
        print(sol)

Output:

{V: 3538, U: 3583, T: 3598, S: 3601, v: 7, u: -3, t: -1, D: 3598, C: 3583, B: 3538, A: 3367, d: -1, c: -3, b: 7, a: 13, M: 2701, y: 13, x: 24}
{V: 3538, U: 3583, T: 3598, S: 3601, v: 7, u: -3, t: -1, D: 3598, C: 28703/8, B: 3538, A: 3367, d: -1, c: 1/4, b: -6, a: 13, M: 2701, y: 13, x: 24}
{V: 3538, U: 3583, T: 3598, S: 3601, v: 7, u: -3, t: -1, D: 3598, C: 114881/32, B: 28379/8, A: 3367, d: -1, c: 27/16, b: -11/2, a: -12, M: 2701, y: 13, x: 24}
{V: 3538, U: 3583, T: 3598, S: 3601, v: 7, u: -3, t: -1, D: 3598, C: 114737/32, B: 28379/8, A: 3367, d: -1, c: -21/16, b: 13/2, a: -12, M: 2701, y: 13, x: 24}