Ibn al-Qayyim (رحمه الله) said:

The scholars dispute two issues. Firstly, if a [Muslim] sneezes, should those who did not [actually] hear him sneeze supplicate for him [i.e. say: “Yarḥamukallāh/May Allāh have mercy upon you”]? There are two opinions, but the most correct opinion is that they should supplicate for him once they confirm that he praised Allāh [i.e. he said: “Al-Ḥamdulilāh”]. The intent here is not for a person to actually hear the sneezing person praising Allāh; rather, the intent is for a person to praise Allāh [after sneezing]. Therefore, once a person verifies that the person who sneezed praised Allāh, this person must supplicate for him. Therefore, if a person sees the motion of a mute person’s lips who just sneezed praising Allāh, he must supplicate for him¹. The Prophet (صلى الله عليه وسلم) said: “Once he praises Allāh [i.e. after sneezing], beseech Allāh’s Mercy for him.”² This is the correct opinion [in relation to this issue].

Finally, should the person who sneezed be reminded to praise Allāh if he forgot to do so? Ibn al-ʿArabī (رحمه الله) said: “He should not be reminded since this is forgetfulness on his part.” Al-Nawawī (رحمه الله) stated: “Whoever claims this is wrong. He must be reminded, since this is a form of advice, and is considered enjoining the good, and is cooperating with each other upon righteousness.” This is [also] Ibrāhīm al-Nakhaʾī’s opinion. However, the Sunnah supports Ibn al-ʿArabī’s opinion, since the Prophet (صلى الله عليه وسلم) never reminded anyone who failed to praises Allāh [after sneezing] to do so let alone to supplicate for his forgiveness, and if reminding him to praises Allāh [after sneezing] was the Sunnah, the Prophet (صلى الله عليه وسلم) would have done this and would have taught others to do so, [yet he did not do this]³. When this person forgot Allāh, Allāh caused the believers’ hearts and tongues to turn away from supplicating for him, and this is a type of punishment.⁴

Endnotes:

[1] This analogy also applies to the deaf person who sees a person’s lips praising Allāh after sneezing. The deaf person must also supplicate for the person who praised Allāh after sneezing, as the deaf person was also able to verify that this person actually praised Allāh after sneezing by reading his lips.
[2] Reported by Muslim (2992).
[3] If reminding the person who failed to praise Allāh after sneezing was from the Dīn, the Messenger (صلى الله عليه وسلم) would have taught us to remind him to praise Allāh, but he (صلى الله عليه وسلم) did not do so; rather, he (صلى الله عليه وسلم) said: “But if he failed to praise Allāh, then do not beseech Allāh’s Mercy for him.” [Reported by Muslim (2992)]

Furthermore, if reminding the person who failed to praise Allāh after sneezing to praise Him was from the Dīn, the Companions would have done it, yet they did not do so, as you will see that Abū Mūsá (رضي الله عنه) never reminded Abū Burdah (رضي الله عنه) to praise Allāh after failing to do so.

Abū Burdah (رضي الله عنه) reported:

I visited Abū Mūsá (رضي الله عنه) while he was in the house of Umm al-Faḍl ibn al-ʿAbbās (رضي الله عنها). I sneezed, but Abū Mūsá (رضي الله عنه) did not beseech Allāh’s Mercy for me. However, when Umm al-Faḍl (رضي الله عنها) sneezed, he beseeched Allāh’s Mercy for her. I informed my mother about this, so when Abū Mūsá (رضي الله عنه) visited my mother, my mother confronted him saying: “My son sneezed, but you did not beseech Allāh’s Mercy for him. However, when Umm al-Faḍl (رضي الله عنها) sneezed, you beseech Allāh’s Mercy for her.” Abū Mūsá (رضي الله عنه) replied: “I heard the Prophet (صلى الله عليه وسلم) say: ‘When one of you sneezes then praises Allāh, you must beseech Allāh’s Mercy for him; however, if he failed to praise Allāh, do not beseech Allāh’s Mercy for him.’ Your son sneezed, but he did not praise Allāh, so I did not beseech Allāh’s Mercy for him. However, when Umm al-Faḍl (رضي الله عنها) sneezed, she praised Allāh, so I beseeched Allāh’s Mercy for her.” My mother [then] said [to Abū Mūsá (رضي الله عنه)]: “You did well.” [Reported by al-Bukhārī in al-Adab al-Mufrad (425), and Shaykh al-Albānī deemed it authentic.]

Therefore, these texts prove that a person should not be reminded to praise Allāh if they forgot to do so after sneezing, since the Prophet (صلى الله عليه وسلم) himself did not do this, nor did he teach the Companions to do this.
[4] Zād al-Maʿād, vol. 2, pg. 403.