error with NIntegrate and RegionPlot

This is a bug in RegionPlot. For a possible workaround, try the following undocumented option

RegionPlot[NIntegrate[PDF[NormalDistribution[0, 1], a], {a, 0, y}] >= 0.2, 
   {x, -1, 1}, {y, 0.1, 0.7}, "NumericalFunction" -> False]

EDIT : Changed for your edited question

$Version

(* "10.2.0 for Mac OS X x86 (64-bit) (July 7, 2015)" *)

Define a helper function that is defined only for numeric arguments

f[y_?NumericQ] :=
  NIntegrate[
   PDF[NormalDistribution[0, 1], a],
   {a, 0, y}];

rgn = ImplicitRegion[
   f[y] >= 0.2 && -1 <= x <= 1 && 0.1 <= y <= 0.7,
   {x, y}];

However, this is very sloo...oow

RegionPlot[rgn, PlotRange -> {{-1, 1}, {0.1, 0.7}}] // 
  AbsoluteTiming // Column

ContourPlot is much, much faster

f[0.62] >= 0.2

(* True *)

ContourPlot[f[y],
   {x, -1, 1}, {y, 0.1, 0.7},
   Contours -> {0.2},
   Epilog -> Text["f[y] \[GreaterEqual] 0.2", {0, 0.62}]] //
  AbsoluteTiming // Column

---

The largest that your integral can be is for y = 0.7

dist = NormalDistribution[0, 1];

Integrate[PDF[dist, a], {a, 0, 0.7}]

(* 0.258036 *)

This is equivalent to

CDF[dist, 0.7] - CDF[dist, 0]

(* 0.258036 *)

Even if you were to integrate from -Infinity, the largest that the integral could be is

Integrate[PDF[dist, a], {a, -Infinity, 0.7}]

(* 0.758036 *)

or equivalently,

CDF[dist, 0.7] - CDF[dist, -Infinity]

(* 0.758036 *)

or more simply

CDF[dist, 0.7]

(* 0.758036 *)

Consequently, since you are looking for the region for which the integral is greater than or equal to 0.95, your region is empty. Note the use of Integrate rather than NIntegrate

RegionPlot[
 Integrate[PDF[NormalDistribution[0, 1], a], {a, 0, y}] >= 0.95, {x, -1, 
  1}, {y, 0.1, 0.7}]

If you reverse the inequality then

RegionPlot[
 Integrate[PDF[dist, a], {a, 0, y}] < 0.95, {x, -1, 1}, {y, 0.1, 0.7}]

Or the same result with

RegionPlot[CDF[dist, y] - CDF[dist, 0] < 0.95, {x, -1, 1}, {y, 0.1, 0.7}]

Try this small variation over your original request

   RegionPlot[
Integrate[PDF[NormalDistribution[0, 1], a], {a, 0, x}] > 0.3 // 
  Evaluate, {x, -10, 10}, {y, -1, 1}]