Does modern Greek still use the six tenses of classical Greek?

No, thank God. Although there’s some noteworthy continuities in what has survived: the morphology and semantics are pretty much the same. In the indicative:

• Present: yes.
• Imperfect: yes. The imperfect shows up in subjunctive contexts, to do the work of the erstwhile optative.
• Aorist: yes.
• Future: no. Replaced by a succession of auxiliary formations (μέλλω, έχω, να subjunctive, θέλω); the θέλω formation has prevailed, and is now a θα particle. Just like the will future in English.
• Perfect: no. Replaced by auxiliary formation, with have or be. Just like the perfect in English.
• Pluperfect: no. Replaced by auxiliary formation, with had or was. Just like the perfect in English.
• Future perfect: no. Replaced by auxiliary formation, with θα and have or be. Just like the future perfect in English.

In the subjunctive, imperative, and participle, the present/aorist distinction survives, and is purely aspectual.