Airfoil variation will make a significant difference in airflow over the wing, thus proper variation according to the lift profile should improve the overall aerodynamic performance. But by how much? Are major aviation manufacturers using it? I can't find any reliable information on this.
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.
Sure the major aircraft companies are using airfoil variation along the wing span. There are a number of considerations in deciding the airfoil sections and their change when it comes to aircraft like its effect on the formation of shock waves (important for transonic airliners), the weight and cost considerations (in designing and manufacturing the wing with different sections). Most of the large aircraft use airfoil variation along the span; exceptions to this are smaller (GA) aircraft, though not all (and smaller rotor blades).
A good source of different airfoils used in aircraft is the University of Illinois Urbana-Champaign Airfoil Coordinates database and also airfoiltools. As an example, airfoil tools lists the airfoils used in the Boeing 737:
Root: Max thickness 15.4% at 19.6% chord; Max camber 0.2% at 5% chord
Midspan: Max thickness 12.5% at 29.7% chord; Max camber 0.8% at 10% chord
Tip: Max thickness 10.8% at 40% chord; Max camber 1.6% at 20% chord
You can see that the thickness reduces as we go out along the span while the camber increases. Basically, this is to optimize weight (thicker spar at root) and aerodynamics performance (thinner airfoils are better at higher altitudes and higher speeds).
Note: Most modern aircraft by large companies (like Boeing and Airbus) use in-house airfoils that are developed specifically for their projects and are confidential information; it is highly doubtful if that will be made available. But with the availability of modern CFD codes and manufacturing techniques (especially composite), we can be sure that the airfoil sections are being varied across the span.