If an aircraft is following a magnetic course changes in magnetic declination as the aircraft moves along its route of flight will affect the true course, and on a long flight such as the hypothetical trip posed here we need to take that into account.
Magnetic declination is empirical data, i.e. there are tables of it, and while formulas exist to approximate it you would normally use the reference tables. There are several sources of data for the worldwide declination of the magnetic field, for example here.
Following the link "Magnetic Model maps and grids" I took D_map_mf_2010.zip which contains a simple lat/lon/declination table (0.5° steps) and wrote a small simulation which moves the aircraft 10 km along the heading, calculates & records the new angular coordinates, moves again and so on. The result is this figure:
The green arrows show the direction of north as displayed by a compass, the blue shows the direction of east.
The black line was generated by the simulation shows the true course of an aircraft with a constant magnetic heading of 90°. While it will fly over north Florida, it deviates markedly to the north over the ocean and just touches the north coast of Spain.
Note that this only accounts for the magnetic declination, since your question assumed no wind.
During a real flight the aircraft may experience unknown crosswinds, so the actual position has to be determined from time to time, and a look into the table / map shows the correct heading at the current position leading to your destination. (If the aircraft only has this map, a compass and a sextant aboard.)
So, if you have this data, you can correct the compass headings such that you fly the true course you want. However, I don't know how it's actually done in aviation.
Here is the code. It is written in python and gnuplot, and you will not need any programming skills to use it.
To answer one question from the comments:
If you carried on around the world, would you return to your starting point? I'm guessing yes as we would be following a contour of equal magnetic potential?
From the physics side, magnetic fields are not conservative and so do not have a potential. However, I think it's clear what you mean.
As long as the heading is +90° or -90°, you will end up where you started. The following figure again shows a travel from KLAX heading 90°, but with a total lengh of 40mio km (i.e. 1000 times the circumfence of the earth and a little more turns for the track as it's not at the equator)
As you can see, there is just a single black track, no deviation.
And if you want to fly to Casablanca, you will need a constant heading of 93,6°. And if you don't land there and have enough fuel, strange things happen:
I would also like to emphasize Bob Jarvis' comment, about how it's done in the real world:
Long distance journeys, whether by air or sea, will generally use a "great circle" where the heading steered is not constant. In practice the continuously changing great circle course is broken up in to a series of segments of constant heading. Reference here.