I'm visualizing a vehicle with a tube that travels the length of the vehicle, top to bottom, going through both propellant tanks. Is this feasible? What are the issues with such a design?
It should be feasible.
The primary issue is going to be temperature; if the tunnel is air-pressurized, going through cryogenic propellant tanks, water will condense and freeze on the inside of the tunnel, and if the tunnel is kept warm enough for crew to move through, it'll be heating the cryo propellants.
If you're not planning to use the tunnel until the propellants are depleted, there's no problem; depressurize the tunnel and it shouldn't have substantial impact on the thermal situation in the tanks.
The Saturn V first stage stacked the liquid oxygen tank above the kerosene tank; the flow rate required by the engines was too large to allow for routing the LOX lines around the outside of the fuel tank, so they had to go straight through. The lines required substantial insulation to keep kerosene from freezing on them, and the routing holes in the fuel tank complicated manufacturing. This is more or less the inverse of the situation with a warm pressurized tunnel going through a cryo fuel tank.
Liquid hydrogen being colder than liquid oxygen, a hydrogen-fueled stage will be more of a thermal engineering challenge than a kerosene-fueled one. There are, naturally, no thermal issues if your propellants are room-temperature storable (e.g. MMH/UDMH/Aerozine-50 with NTO oxidizer).
Depending on the design and purpose of the stage, it might be simpler to put the tunnel alongside the tankage, within an outer structure large enough to contain both, rather than straight through.