As of yesterday (2, May, 2018) a proposal for the budget of 2021-2027 was advanced by the European Commission which refers to this subject (among many others).
Before commenting on this I would like to answer your question directly. The Article 7 of The Lisbon Treaty states the following:
Where a determination under paragraph 2 has been made, the Council,
acting by a qualified majority, may decide to suspend certain of the
rights deriving from the application of the Treaties to the Member
State in question, including the voting rights of the representative
of the government of that Member State in the Council. In doing so,
the Council shall take into account the possible consequences of such
a suspension on the rights and obligations of natural and legal
persons.
Which means that that any suspension of rights (including access to cohesion funds) would be, in theory, possible. In fact the article specifically mentions the possible loss of voting rights, something considered to be extreme by most.
Many nations in the EU, including France and Germany, had already mentioned their intent in procuring a more explicit tie between rule of law and funds. This was discussed at several events including at the Summit of the Southern European countries, in Rome, where the rise of populism was one of the points on the table (EU Observer article):
We support all initiatives aimed at fostering democracy and citizens’
participation. In this respect, we welcome the idea of citizens’
consultations, all over Europe, on core priorities for the future of
the European Union, that could be organised from next spring. In the
same spirit, transnational lists of Members of the European Parliament
to be elected at European level could strengthen the democratic
dimension of the Union.
Considering the joint strength of all nations supporting a reform (the great majority of the EU GDP), it was foreseeable that changes were about to occur. Recently the European Commission proposed a budget for 2021-2027 which states the following:
3. The EU budget and the rule of law: sound financial management
A major innovation in the proposed budget is the strengthened link
between EU funding and the rule of law. Respect for the rule of law is
an essential precondition for sound financial management and effective
EU funding. The Commission is therefore proposing a new mechanism to
protect the EU budget from financial risks linked to generalized
deficiencies regarding the rule of law in the Member States. The new
proposed tools would allow the Union to suspend, reduce or restrict
access to EU funding in a manner proportionate to the nature, gravity
and scope of the rule of law deficiencies. Such a decision would be
proposed by the Commission and adopted by the Council through reverse
qualified majority voting.
Reverse qualified majority voting (RQMV) is basically a qualified majority that needs to be approved by the council after it has been approved by the commission. The implication is that no single nation can boycott the decision of the majority (unlike Article 7 which implies unanimity in the Council).
Officially this decision is not targeting any particular nation. In practice this is widely seen as a reaction to the rise of Populism and possible break of the EU Rule of Law obligations by one or more nations. Jean-Claude Juncker mentioned:
The executive, led by Luxembourg politician Jean-Claude Juncker, wants
to be able to "suspend, reduce or restrict access to EU funding" in a
proportionate manner to protect EU investments and European taxpayers'
money.
"Only an independent judiciary that upholds the rule of law and legal
certainty in all member states can ultimately guarantee that money
from the EU budget is sufficiently protected," the commission said.
...
The move is seen as a way to sanction EU members, notably Poland and
Hungary, two countries which the EU has investigated in recent years
for curbing the judiciary's independence.
Poland is under the commission's Article 7 probe, with negotiations
still ongoing between Brussels and Warsaw on how to come to a
political and legal compromise.
However, commission president Jean-Claude Juncker on Wednesday
insisted that the proposed measure is not against any member state.
"It's not a Polish question, it is a general question," he told
reporters.
So it was not specifically the triggering of Article 7 that led to this (although it might; it's in fact a different process) but the effect was just the same.
You should also be aware that there is already fierce opposition among European MPs to some national leaders, particularly Viktor Orbán. This includes people in its own European political Party, the EPP, which has summoned him quite recently:
...to explain the latest developments related to the Hungarian Higher
Education Act and the national consultation “Let’s stop
Brussels”.
The EPP itself is currently under strong media and popular pressure to expel, or at least react more adequately, to some of the actions of Viktor Orbán.
I think it's safe to speculate that we haven't seen the end of this story yet.
EDIT: Today the EUobserver released an opinion article from Israel Butler mentioning something I was not aware of and that it is of importance for this answer. One of the initial recommendations about relevant actions against authoritarian actions in EU members was actually made by the NGO Civil Liberties Union for Europe (Liberties):
The European Commission has unveiled two new tools to frustrate
aspiring authoritarians in the EU.
The first would allow the commission to block EU funds to governments
that bring independent courts under political control.
The commission has packaged this proposal as a technical precaution,
as previously suggested by the NGO, Civil Liberties Union for Europe.
...
The commission has announced a second tool to be fleshed out at the
end of May: a funding programme on "Justice, Rights and Values", to
strengthen fundamental rights, democracy and the rule of law in the
EU.
This comes in response to a resolution, adopted by a resounding
majority in the European Parliament in April, calling for the
commission to create a "European Values Instrument", also an idea
elaborated by Liberties.
I think reading their (Liberties) paper or even their executive summary might bring some contextualization into the current state of affairs regarding this situation.
EDIT 2: Hungary and Poland to lose quarter of EU funds (EUobserver)
The European Commission has proposed steep cuts in cohesion support in
the next longt-term EU budget to eastern European member states,
according to theEU executive's detailed plans on Tuesday (29 May).
Two countries that have caused headaches for the commission over the
last years over migration and rule of law, Hungary and Poland, will be
hard hit.
Hungary's cohesion allocation will be cut by 24 percent, while Poland
will get 23 percent less money in the 2021-2027 period in 2018 prices.
The overall cut in cohesion policy will be around 10 percent.
The deep cuts in funds to the countries that are the loudest critics
of the EU's migration policy could inflame political tensions among
the eastern and western member states.
The commission however argues that the reduction is not a punishment
for criticism and that it is misleading to compare funds available per
countries, when it calculated the allocation by focusing on regions.
