Yes, it can be ignored. This is the casus irreducilbilis, in which the solutions to a cubic can be expressed only in terms of n-th roots that have complex values. The small imaginary parts are the result of round-off error.
If you'd like to get rid of them and don't mind Root[] objects, specify Cubics -> False on the exact system:
Solve[
Rationalize[Rationalize[
0 ==
10125/(14818 \[Phi]) +
150 y0 (-6308.297007407409` - 148180.` \[Phi]) +
150 (946244.5511111114` + 13000.850814965874`/\[Phi] +
1.11135`*^7 \[Phi] - (
0.022395039313571143` (6959.` + (337500.` -
2250.` y0) \[Phi])^(3/2))/\[Phi])
], 0]
, {y0}, Cubics -> False
] /. Rationalize[\[Phi] -> 0.00145] // N
Solve::nongen: There may be values of the parameters for which some or all solutions are not valid.
(* {{y0 -> -13.9237}, {y0 -> 13.8775}, {y0 -> 1712.28}} *)
Note that in this case Solve[] warns us that not all solutions may be valid, which indeed turned out to be the case. Root[] computes the real roots of a polynomial as Real numbers without complex round-off errors.
If you want a more rigorous method, use Method -> Reduce and increase MaxExtraConditions. Here MaxExtraConditions -> 1 is enough, but I started with MaxExtraConditions -> Infinity:
solRat = Solve[
Rationalize[Rationalize[
0 ==
10125/(14818 \[Phi]) +
150 y0 (-6308.297007407409` - 148180.` \[Phi]) +
150 (946244.5511111114` + 13000.850814965874`/\[Phi] +
1.11135`*^7 \[Phi] - (
0.022395039313571143` (6959.` + (337500.` -
2250.` y0) \[Phi])^(3/2))/\[Phi])
], 0]
, {y0}, Cubics -> False, Method -> Reduce, MaxExtraConditions -> 1
]; (* 7 solutions depending on cases for the parameter phi *)
solPhi =(* select the defined cases *)
Select[% /. Rationalize[\[Phi] -> 0.00145] // RootReduce,
NumericQ[y0 /. #] &];
solPhi // N (* we see the original was correct *)
Solve::useq: The answer found by Solve contains equational condition(s) {0==(192677381803147327908319230731396042083840204210666350+<<22>>)/(246032254379437568608800 (9387804582563199208933776766+<<1>>+138587276555703258030203175 Root[<<1>>&,1])),0==<<1>>/(2<<21>>00<<1>><<1>>),0==<<1>>/(246032254379437568608800<<1>><<1>>)}. A likely reason for this is that the solution set depends on branch cuts of Wolfram Language functions.
(* {{y0 -> -13.9237}, {y0 -> 13.8775}} *)
This time we get more of an idea why the previous Solve[] said not all solutions may be valid. We also see that several other solution-cases were rejected by the OP's original Solve[].
In the OP's second code, increasing MaxExtraConditions yields conditions on the solutions that are helpful in filtering out bogus solutions:
solApprox =
Solve[0 ==
10125/(14818 \[Phi]) +
150 y0 (-6308.297007407409` - 148180.` \[Phi]) +
150 (946244.5511111114` + 13000.850814965874`/\[Phi] +
1.11135`*^7 \[Phi] - (
0.022395039313571143` (6959.` + (337500.` -
2250.` y0) \[Phi])^(3/2))/\[Phi]), y0,
Cubics -> False, MaxExtraConditions -> 1];
solApprox /. \[Phi] -> 0.00145
(* <warnings omitted>
{{y0 -> Undefined}, {y0 -> Undefined}, {y0 -> Undefined}, {y0 -> Undefined},
{y0 -> -13.9237}, {y0 -> 13.8775}, {y0 -> Undefined}}
*)
If we remove the conditions from the solutions with Normal[], we see the last three are the solutions returned by the OP's second Solve[]:
Normal@solApprox /. \[Phi] -> 0.00145
(*
{{y0 -> -68.88}, {y0 -> -67.7773}, {y0 -> 41.1304 - 66.9167 I}, {y0 -> 41.1304 + 66.9167 I},
{y0 -> -13.9237}, {y0 -> 13.8775}, {y0 ->1712.28}}
*)
