Let's analyze the two statements given and compare them with Gause's competitive exclusion principle.

Statement I: "Gause's competitive exclusion principle states that two closely related species competing for different resources cannot coexist indefinitely." This statement is incorrect. Gause's principle actually states that two species competing for the same limited resource cannot coexist indefinitely. If the resources are different, the principle does not necessarily apply because the species are not directly competing against each other for the same needs. Therefore, the correct phrasing should refer to species competing for the same resource.

Statement II: "According to Gause's principle, during competition, the inferior will be eliminated. This may be true if resources are limiting." This statement accurately describes a possible outcome according to Gause's competitive exclusion principle. The principle suggests that in a scenario where two species compete for the same limited resource, one species will often outcompete the other, leading to the decline or possible extinction of the less competitive species.

From the analysis above:

Statement I is false because it incorrectly states that species compete for different resources under Gause's principle.

Statement II is true as it correctly highlights the eventual elimination of one species due to competition over limited resources.

Therefore, the correct answer is:

Option D: Statement I is false but Statement II is true.