First, the Gopala-Purva-Taaapani Sruti(1/2) says, Sri Krishna is ever-young
Gopavesham avraabham TARUNAM kalpadrumasritam.
vayas cha tachchaishavashovayashritam sadaa tathaa jauvanaleelayadritam/ manojna-kaishora-dasha-avalamvitam pratikshanam nutanam nutanam gunair.
In the commentary of the above sloka, Srila Sanatana Goswami writes:
Sri Krishna is ever-young(chira-kishora), ever-beautiful and always without mustache and beard.
In Srimad-Bhagavatam(3/28/17), Sri Kapila says to His mother Devi Devahuti that 'Graceful God is always ever-young'. (Reference: Mahaprabhu Sri Gauranga, Radhagovinda Nath, Sadhana Prakashani, page 146-147).
So Vishnu and His incarnations are always depicted without beard and mustache (probably the only exception is the image at Sri Parthasarathy temple Chennai.)
Other forms of Gods and Goddesses are also mostly depicted as young as that is mentioned in their dhyanamantras. For example, the dhyana-mantra of Devi Durga in Kalika-Purana and Brihat Nandikeswara-Purana and quoted in the Puja-Paddhati book mention Her as 'Nava-yauvana-sampanna'. The complexion, number of arms, the astras etc are almost always explicitly mentioned in the dhyana-mantras.
Devi Dhumavati of the Dasa-Mahavidyas is depiceted as aged as per Her dhyana-mantra. (Reference: Sastramulak Bharatiya Shaktisadhana, Upendrakumar Das,Vol 1, chapter 11. The appendix gives dhyanamantras of Devis).
Brahma is shown with beard in many images probably because He is generally referred to as Grand-father (Pitamaha).
Regarding Hanumanji, I did see different types of images, with and without beard and the mantras composed by Tulsidasji do not mention it clearly so far as I know.