The government needs to take on board these objections to its health reforms

[Theseus]

take on board something (hideous cliche!) means to understand and accept ideas and opinions which may change the way you behave in the future:-
-- Banks need to take on board the views of their customers.
--It seems that young people are finally taking on board the message that it's not cool to smoke. :down:mg:

 

[SBE]

Λαμβάνω (σοβαρά) υπόψη;

 

[Palavra]

For your first example, other than SBE's suggestion, we may say να συνυπολογίσει (συνυπολογίζω). For your second, maybe συνειδητοποιώ (to realise).

 

[Hellegennes]

I think that the phrase implies something more direct than just "take seriously into account". I'd prefer "ενστερνίζομαι".

 

[nickel]

Δίνει την εντύπωση ο ιδιωματισμός ότι σημαίνει υιοθετώ. Να, το λέει σαν δεύτερη σημασία (assimilate, αφομοιώνω) και το OED:

take something on board
informal fully consider or assimilate a new idea or situation: we’ve got to take accusations of sexism on board.

Όμως το παράδειγμα, όπως και τα παραδείγματα του Collins, με κάνουν και μένα να σκέφτομαι «εξετάζω, λαμβάνω (στα) σοβαρά υπόψη»:

to accept (new ideas, situations, theories, etc)
Usage examples

• They sometimes take on board information from other professionals who view the site. - British Medical Journal (2002)
• Nonetheless, we are prepared to listen to the people and take on board their concerns. - Irish Times (2002)
• "I wish my brother would take on board the fact that we need a new employment policy. - Times, Sunday Times (2001)
• "I'm certainly not going to take on board anyone else's problems with who I am," Gordon says. - Rice, Judith, New Testaments (1993)