Stating the obvious:
Given the problem's symmetry, once you have 1 solution, you have 4*3*2=24 solutions.
Kitonum, on my system your loop took several minutes (Maple 15 Windows7x64).
To get data about how long, I rewrote wrapped your code into a procedure:
SolveIt := proc(v::realcons:=7.11,r::range:=0.7..7.11/4,s::realcons:=0.01)
for a from r1 by s to r2 do
for b from a by s to (v-a)/3 do
for c from b by s to (v-a-b)/2 do
for d from c by s to min(3.2,v-a-b-c) do
if delta<s and delta<R then R:=[delta,[a,b,c,d]]: end if:
memory used=25.43GiB, alloc change=0 bytes, cpu time=18.69m, real time=19.85m
[0., [1.20, 1.25, 1.50, 3.16]]
So it took about 20 minutes.
Unfortunately, I'm not competent to make it more efficient.
Note that the procedure above is a little more general in that you can modify the range over which to search (default=0.7..7.11/4) or play around with other values of the sum/product (default=7.11) or of the precision required (default=0.01)
As far as I can tell, your loop rules out multiplicities, i.e. cases where a=b, but it's probably easy to change that by starting the do loops from a-0.01 instead of a, etc. I think.