Former party chair Baroness Warsi has also suggested the Conservatives are in denial about the extent of Islamophobia within its ranks.
The Conservatives have launched an inquiry into all forms of prejudice within the party.
But critics have said it is not independent and would not specifically address anti-Muslim racism.
Ms Dickinson posted the tweet in response to a clip from a speech the MP made in a Commons debate on free school meals.
Ms Shah accused ministers of having “no real understanding, care or emotion” about the extent of child poverty in the UK.
In the clip, the Labour politician – who was born in Bradford and now represents the city – spoke of her experience, being “palmed off” to social services when she was a child, separated from her family and taken on trips to Scarborough, adding “that is what poverty is”.
On her Twitter handle, Ms Dickinson describes herself as a political communications and social media consultant. She is understood to have worked for the Vote Leave campaign during the 2016 Brexit referendum.
The Muslim Council of Britain claimed the party activist had previously shared an Islamophobic conspiracy theory in response to the 2019 terror attack on a mosque in Christchurch, New Zealand, but no action had been taken against her.
In a statement, the group said: “Will this latest blatant racism elicit action? The party must reflect and consider why it chooses to ignore widespread concerns about its institutional Islamophobia.
“If a truly independent inquiry is not enacted with its recommendations implemented, there will be a drip-feed of these stories for a long time to come.”
A Conservative Party spokesman said: “Theodora Dickinson has been suspended pending the outcome of an investigation.”