[This is complementary answer to Justin Tackett answer. I felt somewhat compelled in light of the comments (even after edits), and to illustrate my example (referenced in the answer) with a diagram.]

I say there are 2 naïve situations where the innermost planet is not the closet. In the attached illustration there are only 2 planets and I consider the distance to the Sun (As we might assume the innermost planet is so close to the Sun - so it won't make any difference to take the distance to the Sun instead).

1. In the first example the period (and therefore the semi-major axis) and the pehilion angle of the outer planets are the same. They differ only in eccentricity. We can see by eye that the distance between the planets is always smaller than the distance to the Sun. (Sorry of the size - don't know why so big)

One can justly claim that this position is settings is not possible in real-life - and I'll agree with him. The value in my answer is primarily to demonstrate that there is nothing universal here that can be deduced from the Keplerian system; rather, that every system has its own characteristics.

2. The second example that was given in Justin Tackett answer, includes the two planets have very high eccentricity and about the same semi-major axis and pehilion angle. I admit I thought - in the numbers below given - the effect would be stronger; nevertheless even here it is quite easy to see the average distance between the planets is shorted than the the distance to the Sun. Note that the the diagram is not on scale: the X-axis is longer - had it on scale the orbits would seem much more like strong ellipses. At most of the time the planets are above X>3 while the sun is at 0. this enough to establish I believe my claim here. Moreover, when in opposition the amount of time they are really close is high

[This is complementary answer to Justin Tackett answer. I felt somewhat compelled in light of the comments (even after edits), and to illustrate my example (referenced in the answer) with a diagram.]

I say there are 2 naïve situations where the innermost planet is not the closet. In the attached illustration there are only 2 planets and I consider the distance to the Sun (As we might assume the innermost planet is so close to the Sun - so it won't make any difference to take the distance to the Sun instead).

1. In the first example the period (and therefore the semi-major axis) and the pehilion angle of the outer planets are the same. They differ only in eccentricity. We can see clearly see that the distance between the planets is always shorter than the distance to the Sun. (I used to same orbit phase - but we can allow some small difference - also in the pehilion location)

One can justly claim that this setting is not possible in real-life - and I'll agree with him. The value in my answer is primarily to demonstrate that there is nothing universal here that can be deduced from the Keplerian system; rather, that every system has its own characteristics.

2. The second example that was given in Justin Tackett answer, includes the two planets to have very high eccentricity and about the same semi-major axis and pehilion angle. I admit I thought - in the numbers below given - the effect would be stronger; nevertheless even here it is quite easy to see that the average distance between the planets is shorter than the the distance to the Sun. Note that the diagram is not on scale: the X-axis is longer - had it on scale the orbits would seem much more like strong ellipses. At most of the time the planets are at X>3 while the sun is at 0. this enough to establish I believe my claim here.