Thanks for your response. I found one mistake in my BC, that Up should be in terms of yp in first BC. So, I corrected it but still don't get the result. Kindly check the corrected code.

Subscript[U, 
  p] = (Sqrt[Da]
      E^(-((Y Sqrt[\[Epsilon]])/Sqrt[
      Da])) (-1 + E^((Y Sqrt[\[Epsilon]])/Sqrt[
       Da])) (E^((Sqrt[\[Epsilon]] Subscript[y, p])/Sqrt[
        Da]) (1 + E^((Y Sqrt[\[Epsilon]])/Sqrt[Da])) - 
       2 Da (E^((Y Sqrt[\[Epsilon]])/Sqrt[Da]) - E^((
          Sqrt[\[Epsilon]] Subscript[y, p])/Sqrt[Da])) (-1 + E^((
          Sqrt[\[Epsilon]] Subscript[y, p])/Sqrt[Da])) - 
       2 Sqrt[Da] (E^((Y Sqrt[\[Epsilon]])/Sqrt[Da]) - E^((
          2 Sqrt[\[Epsilon]] Subscript[y, p])/Sqrt[Da])) \[Epsilon]^(
        3/2) + Subscript[y, 
        p] (-2 E^((Sqrt[\[Epsilon]] Subscript[y, p])/Sqrt[Da]) + 
          2 Sqrt[Da] E^((Y Sqrt[\[Epsilon]])/Sqrt[Da]) \[Epsilon]^(
           3/2) - 2 Sqrt[Da] E^((2 Sqrt[\[Epsilon]] Subscript[y, p])/
           Sqrt[Da]) \[Epsilon]^(3/2) + 
          E^((Sqrt[\[Epsilon]] (Y + Subscript[y, p]))/Sqrt[
           Da]) (-2 + Subscript[y, p]) + 
          E^((Sqrt[\[Epsilon]] Subscript[y, p])/Sqrt[Da]) Subscript[y,
            p])))/(2 (-Sqrt[Da] + \[Epsilon]^(
       3/2) - \[Epsilon]^(3/2) Subscript[y, p] + 
       E^((2 Sqrt[\[Epsilon]] Subscript[y, p])/Sqrt[
        Da]) (Sqrt[Da] + \[Epsilon]^(
          3/2) - \[Epsilon]^(3/2) Subscript[y, p])));

Subscript[U, 
  c] = ((-1 + Y) (Sqrt[Da] + Sqrt[Da] Y - 2 Da \[Epsilon]^(3/2) + 
     4 Da E^((Sqrt[\[Epsilon]] Subscript[y, p])/Sqrt[Da]) \[Epsilon]^(
      3/2) - Y \[Epsilon]^(3/2) - 
     2 Sqrt[Da] Subscript[y, p] + \[Epsilon]^(3/2) Subscript[y, p] + 
     Y \[Epsilon]^(3/2) Subscript[y, p] - \[Epsilon]^(3/2) 
\!\(\*SubsuperscriptBox[\(y\), \(p\), \(2\)]\) - 
     E^((2 Sqrt[\[Epsilon]] Subscript[y, p])/Sqrt[
      Da]) (Sqrt[Da] (1 + Y) + (2 Da + Y) \[Epsilon]^(
         3/2) - (2 Sqrt[Da] + (1 + Y) \[Epsilon]^(3/2)) Subscript[y, 
         p] + \[Epsilon]^(3/2) 
\!\(\*SubsuperscriptBox[\(y\), \(p\), \(2\)]\))))/(
  2 (-Sqrt[Da] + \[Epsilon]^(3/2) - \[Epsilon]^(3/2) Subscript[y, p] +
      E^((2 Sqrt[\[Epsilon]] Subscript[y, p])/Sqrt[
      Da]) (Sqrt[Da] + \[Epsilon]^(
        3/2) - \[Epsilon]^(3/2) Subscript[y, p])));

A = FullSimplify[D[Subscript[U, p], {Y, 2}]];

B = FullSimplify[D[Subscript[U, p], {Y, 1}] /. {Y -> 0}];

F = FullSimplify[
   Integrate[Subscript[U, c]/Um, {Y, Subscript[y, p], 1}]];

G = FullSimplify[Subscript[U, p] /. {Y -> Subscript[y, p]}];

FullSimplify[
 DSolve[{X''''[Y] - (Bi*(1 + k))/k X''[Y] - 
     1/(k*Um) (A - Bi*Subscript[U, p]) == 0, 
   Z''''[Y] - (Bi*(1 + k))/k*Z''[Y] + Bi/k*Subscript[U, p]/Um == 0, 
   X''[Subscript[y, p]] == 1/k*G/Um, Z''[Subscript[y, p]] == 0, 
   Z'''[0] - Bi*(Z'[0] - X'[0]) == 0, 
   k*X'''[0] + Bi*(Z'[0] - X'[0]) == B*1/Um, 
   X[Subscript[y, p]] == Z[Subscript[y, p]], 
   F == -k*X'[Subscript[y, p]] - Z'[Subscript[y, p]], X[0] == Z[0], 
   1 == -k*X'[0] - Z'[0] }, {X[Y], Z[Y]}, Y]]