Do we lose significant figures with digital instruments?
As far as I know the visual interpolation of analogous measurements is bad scientific practice anyway.
We assume that rounding errors work in both directions and compensate each other in the long run. On the other hand visual interpolation by a human can introduce systematic errors.

But putting this discussion away one has to note that even if you visually interpolate, you still digitalise your data (just with a finer grid). By digitalising I do not mean that you write them into a computer, but the abstract mapping of continous data points of $\mathbb{R}$ into a finite subset. 

In the end it just is a question of how fine the grid is you are measuring with. And yes, if you don't introduce systematic errors a visual interpolation makes your grid a little bit finer.
On the other hand switching to electronic devices usually gives you orders of magnitude better results.

Some examples:

Let's assume that someone wants to measure lengths of $\mu$m with a normal ruler that has 1 mm spacing. 
It is intuitively clear that this won't work.
(Even if you can interpolate.)

Now let's look at another example that was brought up in the comments by @Zhe. The problem that every number in your computer is an element of a finite set. This introduces rounding errors, when saving real numbers (real as in $\mathbb{R}$).
If you execute the following python code or something similar in your language of choice, you will see this rounding error.

import sys
print(sys.float_info.epsilon)
# for python2:
# print sys.float_info.epsilon


On my 64bit machine this is in the order of $10^{-16}$. The standard deviation of the meter is in the order of $10^{-11}$ (wikipedia).
Floating point noise will rarely be a problem, when dealing with experimental lengths.

This kind of analysation has to be done for every experimental measurement device you are using. Regardless of electronics vs. non electronic measurements. The advantage of "old analogous" measurements is that you can intuitively see from the scale how fine the grid is, that you are working with.