With : t=1;

With[{t = 1}, 
 ListPlot3D[
  Flatten[Table[{x, y, func[\[Xi], \[Eta], t]} /. 
     xyToXiEta[x, y, t], {x, -1, 1, 1/20}, {y, -1, 1, 1/20}], 2], 
  BoxRatios -> {1, 1, 1}, Axes -> True, 
  PlotRange -> {All, All, 2000 {-1, 1}}, AxesLabel -> {x, y, z}]]

Here is an attempt:

func[\[Xi]_, \[Eta]_, 
   t_] = -((1350.` Sech[\[Eta]]^2 Tanh[\[Eta]] (1.` Sech[
             0.1 t - \[Xi]]^2 Tanh[0.1 t - \[Xi]] + 
           0.6` Sech[0.3 t - \[Xi]]^2 Tanh[0.3 t - \[Xi]] + 
           1.6` Sech[0.1 t + \[Xi]]^2 Tanh[0.1 t + \[Xi]]))/((1 - 
           1.5 Sech[\[Eta]]^2) (1 - Sech[0.1 t - \[Xi]]^2 - 
           0.5 Sech[0.3 t - \[Xi]]^2 - 
           1.5 Sech[0.1 t + \[Xi]]^2) (0.3 Sech[\[Eta]]^2 + 
           0.5 Sech[0.1 t - \[Xi]]^2 + 0.3 Sech[0.3 t - \[Xi]]^2 - 
           0.8` Sech[0.1 t + \[Xi]]^2)^2)) // Rationalize // 
   Simplify;
xyToXiEta[x_?NumericQ, y_, t_] := 
  NSolve[{x == \[Xi] - 1/2 Tanh[\[Xi] - 3/10 t] - 
      Tanh[\[Xi] - 1/10 t] - 3/2 Tanh[\[Xi] + 1/10 t], 
    y == \[Eta] - 3/2 Tanh[\[Eta]]}, {\[Xi], \[Eta]}, Reals];
With[{t = 1},
 ListPointPlot3D[
  Flatten[
   Table[{x, y, func[\[Xi], \[Eta], t]} /. xyToXiEta[x, y, t], {x, -1,
      1, 1/20}, {y, -1, 1, 1/20}], 2], BoxRatios -> {1, 1, 1}, 
  Axes -> True, PlotRange -> {All, All, 2000 {-1, 1}}, 
  AxesLabel -> {x, y, z}]]

Unfortunately,

NSolve[{x == \[Xi] - 1/2 Tanh[\[Xi] - 3/10 t] - Tanh[\[Xi] - 1/10 t] - 3/2 Tanh[\[Xi] + 1/10 t], y == \[Eta] - 3/2 Tanh[\[Eta]]}, {\[Xi], \[Eta]}, Reals]

does not function.

When t=0 the function you want to plot is identically zero.