For the case of sixteen diners a four month schedule may be constructed from the five parallel classes of this Answer, using the fifth class to determine the dish assigned to a diner.
That is, let the four months' groupings be as in the first four days or "rounds" of that group tournament question.Then assign the "main" dish (and hosting responsibility) to the first group of the fifth class in January, to the second group of that class in February, to the third group in March, and to the fourth group in April.
Rotate the other three course/dish assignments similarly. For example, in January an appetiser would be brought by those who will bring a main dish in February, a vegetable by those who will bring a main dish in March, and the dessert by those hosting in April.
In this fashion each diner shares exactly one meal with twelve other diners (but no meal with the three diners whose dish assignments agree). Each diner hosts once and is responsible for each type of dish preparation once over the span of four months.
These kinds of arrangements are called resolvable block designs, and while a lot of them have been constructed, you'll find they exist only for special numbers of diners (points) and groups (blocks).
Added:
Let's take a look at the next larger design with similar features,
25 diners with five dish courses and a five month schedule, eating
together in groups of five.
The design uses a complete set of mutually orthogonal latin squares
of order 5, or what is equivalent and convenient for this case, an
orthogonal array (OA) of 25 runs, strength 2, 6 "factors" and index 1.
What this amounts to is a list of twenty-five rows and six columns
in which entries 1 through 5 are distributed so evenly that any two
columns provide all twenty-five possible pairs. We add a letter A
to Y for each participant to each row of a suitable OA library
example (Wayback Machine):
A: 1 1 1 1 1 1
B: 1 2 2 3 4 5
C: 1 3 3 4 5 2
D: 1 4 4 5 2 3
E: 1 5 5 2 3 4
F: 2 1 2 2 2 2
G: 2 2 3 5 1 4
H: 2 3 5 1 4 3
I: 2 4 1 4 3 5
J: 2 5 4 3 5 1
K: 3 1 3 3 3 3
L: 3 2 5 4 2 1
M: 3 3 4 2 1 5
N: 3 4 2 1 5 4
O: 3 5 1 5 4 2
P: 4 1 4 4 4 4
Q: 4 2 1 2 5 3
R: 4 3 2 5 3 1
S: 4 4 5 3 1 2
T: 4 5 3 1 2 5
U: 5 1 5 5 5 5
V: 5 2 4 1 3 2
W: 5 3 1 3 2 4
X: 5 4 3 2 4 1
Y: 5 5 2 4 1 3
The six columns can be used to determine the five month schedule of
dining partners and the rotation of dish/course assignments however
we like. For example, we can interpret the first five columns as
defining which groups of five eat together in the respective five
months of the schedule. That leaves the sixth and final column for
use in determining what dish each person brings. Add the entry of
the sixth column to the number of the month and reduce mod 5:
Course: (column(6) + # of month) mod 5
main 0
appetiser 1
salad 2
vegetable 3
dessert 4
The existence of such "complete" designs is known only for orders
that are powers of primes, e.g. the cases 4 and 5 discussed above,
which in turn require the square of that order in participants or
"runs" to use the OA term. It is a famous conjecture that these
are the only such possibilities. Stated more precisely, we'd like
to know if all "projective planes" are of prime power orders.
Further Reading:
Mutually orthogonal latin squares
Orthogonal array
Projective plane
