Tell MATLAB not to update the next default color for a specific plot

One solution that doesn't depend on any undocumented features is to use the more primitive line function to produce some of your plots instead of the higher-level plot function. You should get the result you want if you plot the third line with line, which doesn't increment the color order index used by plot:

hold on;
plot(rand(1, 20));
hp = plot(rand(1, 10)+1);
line(11:20, rand(1, 10)+1, 'Color', get(hp, 'Color'), 'LineStyle', '--');
plot(rand(1, 20)+2);

When you plot on an axes an undocumented axes variable is used to control what value in the color order is used, I've not explored them much so you will need to explore a bit more in-depth to fully understand how they work - but in essence they are:

ax = axes();
ax.ColorOrderIndex_I
ax.ColorOrderMode

Updating your example (and data a little - as I found it easier to view) - you can take one away from the ColorOrderIndex_I after you have plotted the "two lines as one":

v1 = ones(20,1);
v2 = v1(1:10)+1;
v3 = v1+2;
figure;

ax = subplot ( 2, 1, 1 )
hold on 
plot(v1);
plot(v2);
plot(v3);
ax = subplot ( 2, 1, 2 );
hold on 
plot(v1);
pl=plot(v2);
plot(11:20,v2,'color',get(pl,'color')   ,'LineStyle','--'); 
ax.ColorOrderIndex_I = ax.ColorOrderIndex_I-1;
plot(v3); 

Note: Using undocumented feature - tested r2015b.

LuisMendo's comment works well, so I put it into a function:

function undoColorOrderUpdate(axis, steps)
    if ~exist('axis', 'var')
        axis = gca;
    end
    if ~exist('steps', 'var')
        steps = 1;
    end

    oldindex = get(axis, 'ColorOrderIndex');
    numcolors = size(get(axis, 'ColorOrder'),1);
    newindex = mod(oldindex-1-steps, numcolors)+1;
    set(axis, 'ColorOrderIndex', newindex);
end

You can then put undoColorOrderUpdate(); or undoColorOrderUpdate(gca, 1); before or after your to-be-ignored plot. If you put it before, you neither need to use a handle nor set the color manually anymore:

hold on;
plot(rand(1,20));
plot(rand(1,10));
undoColorOrderUpdate();
plot(11:20,rand(1,10),'LineStyle','--');
plot(rand(1,20));