Do airliners like Boeing and Airbus use airfoil variation along the span for their wings?
Every good design varies at least the thickness: While the root needs to be thicker to enclose an efficient spar, the tip can be optimised for aerodynamics, which usually results in a root thickness of 16% to 21% and a tip thickness between 9% and 13%. Look at Dave Lednicer's incomplete guide to airfoil usage for an exhaustive collection of aircraft and their respective root and tip airfoils. Generally, larger aircraft tend to use thicker airfoils because scaling laws dictate that their structure has to carry higher loads.

Transsonic designs like airliners need to pay special attention to thickness because a thicker airfoil will encounter local supersonic flow at lower flight Mach numbers. Also, a thicker wing will store more fuel and will be lighter, so the compromise is around 13% thickness on average. With supercritical airfoils, a root thickness of 15% and a tip thickness of 11% has become widely used. Unfortunately, the exact airfoils of modern airliners are not published.

Another reason for varying airfoils is wing sweep. Left with the same airfoil over span, the isobars on the surface would show less sweep, creating a steep pressure recovery at the center wing (with potential for early stall) and suction peaks near the leading edge at the tips. Such an isobar pattern will use the available wing area less efficiently, so designers strive to make the isobars parallel to the local sweep angle by adjusting the local airfoils. At the root the maximum thickness must be shifted forward while tip airfoils need a more backward maximum thickness. Also, camber near the root is reduced up to the point where negative camber along the mid section of the airfoil chord is used.

Here are a few older ones:

Type                        Root airfoil           Tip airfoil
Boeing 377 Stratocruiser    Boeing 117 (22%)       Boeing 117 (9%)
Douglas DC-7                NACA 23016 (16%)       NACA 23012 (12%)
Lockheed 749 Constellation  NACA 23018 (18%)       NACA 4412 (12%)
Lockheed 385 L-1011 TriStar  ? 12.4%                ? 9%
Saab 340                    NASA MS(1)-0316 (16%)  NASA MS(1)-0312 (12%)
Tupolev Tu-104              PR-1-10S-9 (15.7%)     PR-1-10S-9 (12%)
Vickers VC-10               Pearcey 13%            Pearcey 9.75%
VFW 614                     NACA 63A015 (15%)      NACA 65A012 (12%)


Modern CFD techniques and composite structures allow to optimise the local contour at the wing-fuselage intersection such that no single root airfoil is used but a continuous variation which delays local flow separation and shock strength as much as possible. This also means that no stock airfoil is used, and most modern designs have their custom-made airfoils.