The question formulated in the title is answered in the case of a separable algebra. The necessary and sufficient condition in this case is that the ideal in question is not finitely (algebraically) generated. We conjecture that this result is true in the general case.
"When does a closed ideal of a commutative unital Banach algebra contain a dense subideal?." Funct. Approx. Comment. Math. 44 (2) 285 - 287, June 2011. https://doi.org/10.7169/facm/1308749132