Wikipedia states that Vanguard I is the oldest artificial satellite still in orbit. What is the oldest that's still in use?
GOES 3 is most likely the oldest satellite in operation as of early 2014. Launched in 16 June 1978 as a weather forecasting satellite, it was repurposed as a communications satellite when it became unusable for meteorological studies in 1989.
The first satellites listed by SATCAT with a status of + (operational) are CALSPHERE 1 & 2, passive spherical surveillance calibration targets built by the Naval Research Laboratory, first one launched in December of 1962, but apparently no longer in use. LCS 1, a hollow metal sphere with a precisely defined radar cross-section launched on the 6th of May 1965, is still used (along with LCS 4, 7th Aug 1971) to calibrate ground-based radars.
The oldest still operational communication satellite in use is the low budget amateur radio satellite AMSAT-OSCAR 7 made by radio amateurs.
It was launched on 15th of November 1974 from Vandenberg Air Force Base with a Delta 2000 rocket. OSCAR 7 has operational HF/VHF/UHF transponders, which allows communication over distances of up to 9,000 km with a relatively simple ground station equipment.
The satellite is in a 1,500 km retrograde polar orbit. The batteries are dead, although the satellite's solar cells works fine. The satellite is used daily during times when the solar cells are sunlit.
What is the oldest that's still in use?
Depending on what "still in use" means, I would venture that it is LAGEOS-1, launched on 4 May 1976. That's two years before GOES-3, and over a year before the Voyager spacecraft. The LAGEOS satellites are simple to the extreme. They are balls of solid brass covered with retroreflectors, and that's it. No sensors, no effectors, no attitude control, no comm, no avionics.
Researchers ping those retroreflectors with ground-based lasers. That was and remains the intended use of these satellites. The returns from those pings tells researchers range, range rate, and bearing. From that, the orbits of the LAGEOS satellites can be determined with very high precision. From those very precise orbit determinations, LAGEOS provides insight on diverse topics such as the validity of general relativity, the shape of the Earth and the distribution of mass within the Earth, and plate tectonics.
The LAGEOS satellites will cease to serve their initial function when they deorbit (that's million of years from now), when their retroreflectors degrade to such an extent that they can't be seen (that's also a long time), or when they are smashed by another satellite or by orbital debris.