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When a person sneezes while the Friday sermon (khutba) is being delivered, the rulings regarding saying ‘Alhamdulillah’ and responding with Yarhamukallah are as follows:
Saying Alhamdulillah: The person who sneezes should say Alhamdulillah (Praise be to Allah) quietly to himself (silently). He should not raise his voice. 

Saying Yarhamukallah: If someone hears him sneeze and praise Allah, it is not permissible for that person to respond out loud with Yarhamukallah (May Allah have mercy on you). Doing so constitutes forbidden speech during the khutba. 
Scholarly Views Across the Madhhabs

The major schools of Islamic jurisprudence (Hanafi, Maliki, Shafi’I, and Hanbali) agree that listening attentively to the khutba and maintaining absolute silence takes precedence over standard social etiquettes like tashmit (responding to a sneeze).
Imam Malik stated in Al-Mudawwanah that if someone sneezes during the khutba, he should praise Allah quietly to himself, and no one should say Yarhamukallah back to him. 

Other scholars emphasize that while praising Allah is inherently legislated, performing it out loud or answering it during the sermon disrupts the mandatory silence of the khutba and turns it into prohibited speech (laghw). 
Daleel (Evidence)

The legal reasoning is derived from the general texts prohibiting speech during the sermon, alongside specific prophetic warnings regarding distractions:

The Prophet صلى الله عليه وسلم strictly commanded silence during the Friday sermon. Abu Hurairah reported that the Messenger of Allahﷺsaid:
“If you say to your companion ‘Listen’ on Friday while the Imam is delivering the khutba, you have engaged in idle talk.” (Sahih al-Bukhari and Sahih Muslim)

If saying something as noble as commanding good, like saying (“Listen”) invalidates the reward of the Friday prayer due to breaking the silence, then responding to a sneeze or speaking out loud carries the same or greater prohibition.

While responding to a sneeze is normally a stressed Sunnah outside of prayer and sermons, listening to the khutba is a binding obligation (wajib) upon those attending Jumu’ah. In Islamic legal theory, an obligatory duty to listen takes precedence over a non-obligatory or conditional Sunnah response when the two conflict in timing. And Allah knows best. Imam Abu Laith Luqman Ahmad