Al-Allaamah al-Faqeeh Muhammad ibn Saalih al-Uthaymeen (rahimahullah) in explanation of aayatul imtihaan:

‎قُلْ إِن كُنتُمْ تُحِبُّونَ اللَّهَ فَاتَّبِعُونِي يُحْبِبْكُمُ اللَّهُ وَيَغْفِرْ لَكُمْ ذُنُوبَكُمْ وَاللَّهُ غَفُورٌ رَّحِيمٌ

Say (O Muhammad) if you (truly) love Allah, then follow me, Allah will love you and forgive you your sins. And Allah is Oft-Forgiving, Most Merciful. (Aali Imraan: 31)

This verse is named the aayatul mihna, meaning the verse of examination, because a people claimed that they loved Allah, so Allah said to the Messenger,

قُلْ إِن كُنتُمْ تُحِبُّونَ اللَّهَ

Say (O Muhammad) if you (truly) love Allah

This is the sign of a persons love for his Lord, that they follow the Messenger (peace and blessings upon him) and everyone who is more ardent in following the Messenger then he is more beloved to Allah, however lets ponder and look at the result of loving Allah; Allah did not command the Messenger to say “If you love Allah, follow me and be truthful in that” rather He said

يُحْبِبْكُمُ اللَّهُ

Allah will love you

and this is the tremendous fruit and result which every person seeks. Hence the reality is that you loving Allah is not the same as you being loved by Allah. Due to this He said,

فَاتَّبِعُونِي يُحْبِبْكُمُ اللَّهُ

then follow me, Allah will love you

Therefore if you were to seek after the Love of Allah and you love Him, then it is upon you to follow the Messenger (peace and blessings of Allah upon him).

Is it possible to understand from this verse that the actions of the people of innovation belies their claim of loving the messenger (peace and blessings upon him)?

The answer to this, is yes, because we say, if you truly loved the messenger, you would indeed follow him, and if you truly loved Allah, you would definitely follow the Messenger, and if you truly took him (peace and blessings of Allah upon him) as a leader you would not precede him (i.e. utter statements or perform actions which he did not legislate), and how can he be your leader in reality whilst you oppose him, so where is the following of this leadership?

And due to this blatant opposition, how easy it is to destroy their claims; those who say, you don’t celebrate the birthday of the Prophet, and you don’t perform long extended poetic salaat and salaam and praise upon him,  therefore you don’t love him. So we say in reply to them, Glory be to Allah! Which of us has more right to loving the Messenger?! Those who follow his Sunnah and do not leave it, or those who perform that which he (peace and blessings of Allah upon him) warned against of innovations?! There is no doubt that those who truly love Allah is the first type, and destroying their claims does not require a strong axe, rather it is very easy to destroy.

Paraphrased from Sharh Iqtida as-Siratul Mustaqeem pg 723-724